Method and apparatus for optically measuring periodic structures using orthogonal azimuthal sample orientations

ABSTRACT

An optical metrology apparatus for measuring periodic structures using multiple incident azimuthal (phi) and polar (theta) incident angles is described. One embodiment provides the enhanced calculation speed for the special case of phi=90 incidence for 1-D (line and space) structures, which has the incident plane parallel to the grating lines, as opposed to the phi=0 classical mounting, which has incident plane perpendicular to the grating lines. The enhancement reduces the computation time of the phi=90 case to the same order as the corresponding phi=0 case, and in some cases the phi=90 case can be significantly faster. One advantageous configuration consists of two measurements for each sample structure, one perpendicular to the grating lines and one parallel. This provides additional information about the structure, equivalent to two simultaneous angles of incidence, without excessive increase in computation time. Alternately, in cases where the computation for phi=90 is faster than the corresponding phi=0 incidence, it may be advantageous to measure parallel to the grating lines only. In the case where two sets of incident angles are used, the incident light can be polarized to provide a total of four sets of data—R s   0 , R p   0 , R s   90 , R p   90 —for each incident polar angle, all from the same structure.

This application claims priority to Provisional Patent Application No. 60/872,010 filed Nov. 30, 2006; the disclosure of which is expressly incorporated herein by reference in its entirety.

TECHNICAL FIELD OF THE INVENTION

This invention relates to an optical metrology apparatus and methods and systems for measuring periodic structures using multiple incident azimuthal (phi) and polar (theta) angles, and particularly to enhanced calculation speed for a special case of phi=90 incidence for 1-D (line and space) gratings, having an incident plane parallel to grating lines. This results in additional datasets to supplement data collected in the phi=0 classical mount configuration without an untenable increase in computation cost.

BACKGROUND OF THE INVENTION

A widely referenced source on a rigorous coupled wave (RCW) algorithm is that of Moharam and Gaylord (M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, J. Opt. Soc., Am. A, Vol. 12, No. 5, p. 1068 (1995)). A schematic from their paper showing the diffraction problem is reproduced in FIG. 1. In particular, FIG. 1 defines the various incident conditions for the diffraction problem. The plane of incidence is defined by the polar angle, theta, and the azimuthal angle, phi. The azimuthal angle defines the angle the incident plane makes with the plane perpendicular to the grating lines, so that phi=0 corresponds to the classical incidence case. The angle psi defines the direction of the electric field with respect to the plane of incidence, with psi=90 corresponding to s polarized and psi=0 to p polarized incident light.

As shown in FIG. 1, a diffraction grating 100 has a grating region 104 formed in a substrate 102 (the substrate being designated as Region II). The grating region 104 has a height d as indicated in the figure. Region I is comprised of the material above the grating region, in this case in air or vacuum space. As indicated in FIG. 1, the grating region may be formed of alternating grating lines 106 and grating spaces 108. The grating lines 106 may have a width 108. The grating periodicity is characterized by the grating period 10 as indicated.

It will be recognized that a diffraction grating may be formed in other manners than that of FIG. 1 and that FIG. 1 is only one exemplary diffraction grating as known to those skilled in the art. For example, a diffraction grating need not be formed utilizing spaces. FIG. 1B shows one such alternative diffraction grating. As shown in FIG. 1A, the diffraction grating 100 may be comprised of grating lines 106A and 106B, again having a grating region 104 with height d. In this example, grating lines 106A and 106B will be formed in a manner in which the lines have different optical properties. Further, though the examples shown include gratings having two different optical properties within each period of the diffraction grating, it will be recognized that the diffraction grating may comprise three or more different materials within each period. Likewise, though each grating line is shown as a single material, it will be recognized that the grating lines may be formed of multiple layers of the same or different materials. In addition, though the grating lines are shown as being “squared off,” it will be recognized that each line may have sloped sides, curved edges, etc.

With reference again to FIG. 1, an x-y-z coordinate system is shown having a frame of reference in which the x-direction is shown as being perpendicular to the original alignment of the grating lines. The plane of incidence 112 of the incident light is defined by the polar angle 114, theta and the azimuthal angle 116, phi. The electric field 120 has a propagation vector 122 (k) of the incident wave. The unit vectors 124 (t) and 126 (n) are tangent and normal to the plane of incidence, respectively. As mentioned above, the angle 128, psi, defines the direction of the electric field with respect to the plane of incidence.

The RCW method involves the expansion of the field components inside and outside the grating region in terms of generalized Fourier series. The method consists of two major parts—an eigen-problem to determine a general solution inside the grating layer, and a boundary problem to determine the reflected and transmitted diffracted amplitudes along with the specific solution for the fields inside the grating region. The Fourier series are truncated after a finite number of terms. The truncation is usually characterized by the truncation order, N, which means that 2N+1 spatial harmonics are retained in the series (positive and negative terms to +N, and the 0 term).

Standard methods for solving the eigen-problem, boundary problem, and the various other matrix multiplications and inversions involved are order N³ operations. This means that an increase of the truncation order by a factor of two results in an increase in overall computation time by a factor of approximately 8. The truncation order required for convergence is determined by the specifics of the diffraction problem, and generally increases for larger pitch to incident wavelength ratios and larger optical contrast between grating lines and spaces. The result is that while some diffraction problems are very tenable, others quickly become impractical to solve due to a large computation cost.

In the case of the phi=0 classical mount, the diffraction problem decouples into TE and TM components, which can be solved separately (for the phi=0 mount, TE polarization corresponds to s polarized incident light, and TM polarization corresponds to p polarized incident light). Any arbitrary polarization is decomposed into a combination of the TE and TM problems. In practice, the incident light is often purely TE or TM polarized, and only one case needs to be solved. For given truncation order N and classical mount the eigen-problem is of size 2N+1, and the boundary problem is of size 2(2N+1).

The general case where phi≠0 is known as conical diffraction. In this case, the s and p components are coupled, with a corresponding increase in the amount of computation time. The boundary problem involves 4(2N+1) sized matrices. The eigen-problem has been successfully decoupled into two smaller eigen-problems, each of size 2N+1 (see Moharam and Gaylord 1995 referenced above, or S. Peng and G. M. Morris, J. Opt. Soc. Am. A, Vol. 12, No. 5, p. 1087 (1995)). Therefore, the computation time for the general conical incidence case suffers a factor of 2 increase for the eigen-problem and a factor of 8 increase for the boundary problem compared to the corresponding classical mount case with same polar incident angle, theta.

Analysis of a diffraction grating problem is of particular use to determining the various characteristics of the diffraction grating structure. For example, critical dimensions of a device (such as in semiconductor processing in one exemplary use) may be monitored by evaluating the characteristics of a diffraction grating as is known in the art. By evaluating data from known optical metrology tools using regression and/or library methods, the diffraction analysis may lead to, for example, a determination of the grating line widths, the grating height/depth, the period of the grating, the slopes and profiles of the grating, the material composition of the grating, etc. As known in the art, such grating characteristics may be related to the characteristics of a device that is being analyzed, such as for example but not limited to widths, heights, depths, profiles, etc. of transistors, metallization lines, trenches, dielectric layers, or the like, all as is known to those skilled in the art. Since the regression and/or library methods may require many calculations of diffraction efficiencies, special consideration must be given to computation expense in such applications.

SUMMARY OF THE INVENTION

An optical metrology apparatus for measuring periodic structures using multiple incident azimuthal (phi) and polar (theta) incident angles is described. One embodiment provides the enhanced calculation speed for the special case of phi=90 incidence for 1-D (line and space) structures, which has the incident plane parallel to the grating lines, as opposed to the phi=0 classical mounting, which has incident plane perpendicular to the grating lines. The enhancement reduces the computation time of the phi=90 case to the same order as the corresponding phi=0 case, and in some cases the phi=90 case can be significantly faster. One advantageous configuration consists of two measurements for each sample structure, one perpendicular to the grating lines and one parallel. This provides additional information about the structure, equivalent to two simultaneous angles of incidence, without excessive increase in computation time. Alternately, in cases where the computation for phi=90 is faster than the corresponding phi=0 incidence, it may be advantageous to measure parallel to the grating lines only. In the case where two sets of incident angles are used, the incident light can be polarized to provide a total of four sets of data—R_(s) ⁰, R_(p) ⁰, R_(s) ⁹⁰, R_(p) ⁹⁰—for each incident polar angle, all from the same structure (R_(s) ⁰ being a data set having incident phi=0 and incident polarization normal to the plane of incidence, R_(p) ⁹⁰ being a data set having incident phi=90 and incident polarization within the plane of incidence, etc.).

In one embodiment, the techniques described herein provide an optical metrology apparatus and methods and systems for measuring periodic structures using multiple incident azimuthal (phi) and polar (theta) angles, and particularly to enhanced calculation speed for a special case of phi=90 incidence for 1-D (line and space) structures, having an incident plane parallel to grating lines.

In one embodiment, a method of reducing an RCW calculation for the phi=90 incidence mount by exploiting the degeneracy in the resulting diffraction problem is provided. The method may include using this calculation in the regression part of an optical grating measurement. Alternately, the method may include using the enhanced speed in the generation of a database library to be used in conjunction with an optical grating measurement. The optical method could be any in existence, such as reflectometry, polarized reflectometry, ellipsometry, polarimetry, etc. and can be broadband or single wavelength.

The method may include illuminating a grating structure with polarized or unpolarized, monochromatic or broadband light. The incident light may be at one or more polar angles, theta, at the phi=0 and phi=90 azimuthal directions for each of the polar angles. The method may further include detecting the response, for a total of up to four datasets per grating sample per incident polar angle, which are then simultaneously analyzed in order to take advantage of the enhanced information content contained in the multiple datasets. The calculation time is reduced compared to conventional RCW formulations due to the reduced calculation requirements for the phi=90 cases.

One or more diffracted orders may be detected along with or instead of the 0'th order. Further when the detected response is reflected or diffracted intensity, one or more of the datasets may be used to normalize the other datasets, making an absolute calibration of the tool unnecessary.

One or more of the datasets may be used to normalize the other datasets, making an absolute calibration of the optical tool unnecessary, and the inverse ratio is substituted in calculations for specific wavelength regions where the denominator of the original ratio is near zero.

One or more of the datasets may be used to normalize the other datasets, making an absolute calibration of the tool unnecessary, and the inverse ratio is substituted in calculations for specific wavelength regions where the denominator of the original ratio is near zero, and a weighting function is used to equalize the contribution to the merit function regardless of reflectance ratio magnitude.

In addition, one or more of the datasets may be used to normalize the other datasets, making an absolute calibration of the optical tool unnecessary, and the data regions where the denominator of the ratio is near zero are dropped from the analysis.

Data collected from the diffracting structure may be normalized by data from a nearby uniform film structure having the same stack layer structure as the diffracting structure.

The angle of incidence may be explicitly varied by changing the polar angle of incidence of the optical plane, or by rotating the optical plane or sample at fixed polar angle to generate phi=0 and phi=90 incident data.

The multiple angle of incidence data may be generated through use of a high numerical aperture optic (so it contains a spread of angles) and selecting specific angles using an aperture stop to allow light incident at only specific angles.

In addition, multiple angles of incidence may be allowed, and the data simultaneously analyzed.

Further, the method may include only measuring and analyzing the phi=90 incidence data.

As described below, other features and variations can be implemented, if desired, and a related method can be utilized, as well.

DESCRIPTION OF THE DRAWINGS

It is noted that the appended drawings illustrate only exemplary embodiments of the invention and are, therefore, not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments.

FIG. 1 shows a schematic diagram of diffraction problem illustrating polar (theta) and azimuthal (phi) incident angles.

FIG. 1B shows an exemplary alternative diffraction grating.

FIGS. 2A AND 2B shows rotation of the stage/sample/objective by 90 degrees and back for each measurement.

FIGS. 3A-3E illustrate using a high NA objective with an aperture stop that allows light incident at specific angles. Multiple incident angles are achieved by rotating the sample, aperture, or objective. One modification might simultaneously illuminate the grating from both directions (phi=0 and phi=90) as shown in FIG. 3D and FIG. 3E.

FIG. 4 illustrates an apparatus for collecting VUV-Vis reflectance data at multiple incident angles.

DETAILED DESCRIPTION OF THE INVENTION

One way to directly attack the time required for the RCW method is to reduce the number of spatial harmonics involved in the eigen-problem, the boundary problem, or both. This was done for the general case to a large extent in the work of Moharam and Gaylord in their papers: M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068-1076 (1995) and M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077-1086 (1995). Subsequent modifications to enhance the TM convergence have included those techniques shown in: P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779-784 (1996); G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019-1023 (1996); and L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996). However, for certain incidence conditions where the incident plane wave has an x-periodicity that is the same as or is a multiple of the grating period, the diffraction problem benefits from additional degeneracy, and the coupled equations can be even further reduced.

One case where this occurs is when the plane of incidence is in the phi=90 mount so that the incident wave is constant with x, and the resulting degeneracy can be exploited to reduce the total number of unknowns. This reduces the total number of spatial harmonics required for the conical diffraction case from 4(2N+1) to 4(N+1). The result is a reduction of the computation time required by a factor of approximately 8 compared to the standard phi=90 conical diffraction case. In addition, a small but significant reduction in computation time compared to the classical mount with same theta is achieved. There are still two eigen-problems, but each of size N+1, leading to an overall reduction of the eigen-problem by a factor of approximately 4 compared to the comparable phi=0 case. The boundary problems for the two cases require approximately the same computation time (2(2N+1) vs. 4(N+1) matrix sizes). Therefore, the computation time for the phi=90 case is reduced to the same order as the corresponding phi=0 case. The phi=90 case can sometimes be significantly faster than the phi=0 case, depending on how much influence the eigen-problem has on the overall computation time. The result is that the additional azimuthal angle phi=90 can be added to the data without an excessive increase in computation time. It will be recognized that although the concepts described herein may refer to analysis at particular angles such as phi=0 and phi=90, the concepts are not limited to use of these exact angles. For example, equipment and sample tolerances may result in other angles being actually used as variability from an anticipated angle is to be expected in real world applications. In addition, variations from the angles of best choice may be purposefully allowed beyond such tolerances while still obtaining the benefits of the techniques described herein. For example, the optical system and grating may be of such a nature that variations from the most desirable angles will still provide a sufficiently accurate calculation at other angles such that the data collected and the calculations may be effectively similar to the use of phi=0 and phi=90 to the extent that sufficient accuracy for a particular application is obtained. In this manner angles that deviate from the phi=0 and phi=90 may be considered to be effectively phi=0 and phi=90 for the purpose of utilizing a metrology tool implementing the techniques described herein for a given application.

An effective way of increasing the amount of information that can be extracted from a single sample is to collect more data sets. This is often done by using multiple angles of incidence theta or, in the case of grating structures, using multiple azimuthal angles. For instance, T. Novikova, A. De Martino, S. B. Hatit, and B. Drevillon, “Application of Mueller polarimetry in conical diffraction for critical dimension measurements in microelectronics,” Appl. Opt. 45, 3688-3697 (2006) shows a technique to extract more information about grating line-shapes using multiple azimuthal angles in conjunction with Mueller polarimetry.

This is significant since the hardware implementation for multiple phi configurations is considerably simpler than cases where multiple polar angles are used. In the simplest case, a fixed angle theta can be used, while the sample or stage is rotated through the azimuths. Alternately, the optic objective could be rotated. To take advantage of the above mentioned RCW improvements, the system would simply have to rotate the stage/sample/objective by 90 degrees and back for each measurement (FIGS. 2A-2B), generating two datasets for each sample, at orthogonal incident conditions.

As shown in FIG. 2A, a substrate is provided with a diffraction grating 100. A source 206 provides an incident light wave 200 at a polar angle 114 (theta) and an azimuthal angle 116 (phi). As shown in FIG. 2A phi=0 so no angle is indicated. A detector 208 may be provided to detect the diffracted orders of reflected light 202. As used herein, light detected from a diffraction grating may be referred to as reflected light, reflected data, or the like and may include one or both of specular reflection of the zero order light and higher order diffracted light.

Though not shown in the figures provided herein, a computer system, processor, or the like may be coupled to the detector to process the collected data according to the analysis techniques described herein. The rotation of the diffraction grating with reference to the position of FIG. 2A is shown in FIG. 2B. As shown in FIG. 2B, the diffraction grating has been rotated (for example by rotating the stage, the sample, or the incident light or a combination thereof) so as to create an azimuthal angle 116 (phi) that is phi=90.

Another means for collecting data at multiple angles of incidence could utilize the large cone angle inherent in high numeric aperture (NA) normal incident objective systems in conjunction with an aperture stop to allow light incident at specific angles, as shown in FIGS. 3A-C. One advantage of this configuration is that multiple polar angles can be more easily incorporated, in addition to the orthogonal azimuthal angles. As shown in FIG. 3A an aperture 300 provides collimated light to an objective 302 which focuses the light on a diffraction grating 100 at a polar angle 114 theta. Different aperture settings are shown between FIGS. 3A and 3B to illustrate utilizing different polar angles theta. As shown in FIGS. 3A and 3B, the incident wave is not rotated from the classical phi=0 orientation. FIG. 3C illustrates a rotation such that phi does not equal zero, for example phi=90.

Another possibility might simultaneously illuminate the sample from both directions (phi=0 and phi=90). This could be implemented using two separate, orthogonal optic planes, or a modified version of the high NA system (FIG. 3D). The configuration of FIG. 3D is similar to that of FIG. 3A except a modified aperture 300A is provided. A top view of the modified aperture 300A is shown in FIG. 3E. As can be seen, the aperture 300A allows for light to pass in multiple directions, more particularly in a manner such that phi=0 and phi=90 may be simultaneously illuminated.

It will be recognized by those in the art that the embodiments of FIGS. 2A-2B and 3A-3D are merely illustrative so as to demonstrate the technique of changing the azimuthal angle. A wide range of optical metrology tool configurations may be utilized to achieve the desired incidence conditions so as to acquire the data sets described herein. These data sets may then be utilized in a manner that yields desired characteristics of a diffraction grating from the reduced computational complexity analysis that is achieved by collecting the rotated data as described in more detail below. It will also be recognized that although the present disclosure generally is described in reference to data collected at two phi angles (phi=0 and 90 angles), the techniques described herein are not limited to such techniques. In particular, the techniques described herein provide a reduced computation technique that may be utilized for measurements collected at a single phi angle. Further, additional phi angles beyond two may also be utilized to collect data while still obtaining the benefits described herein. Thus, the concepts described herein are not merely limited to collection of data at two phi angles.

Note that collecting multiple broadband datasets using different azimuthal incidence conditions is distinct from the old method of “phi scatterometry”, where the entire dataset consists of the diffraction spectrum of a single wavelength as a function of the azimuthal angle, phi.

The light can additionally, be polarized parallel and perpendicular to the plane of incidence, so that four simultaneous data sets per incident theta can be obtained for a given 1-D grating structure, with corresponding enhancement in information. Alternately, each of the four configurations can be explored for a particular structure, and the most promising configuration employed in practice for measuring that particular structure.

Further, the additional datasets may enhance the information content to the extent that fewer wavelengths in a broadband system can be used in the analysis, and in this way the measurement can actually be made faster than a single angle of incidence configuration broadband measurement. In other-words, the total number of calculations with respect to incident conditions, including wavelength and angle, may be reduced over that required for a single incident condition over many more wavelengths, while still extracting the same information about the grating structure.

Another advantageous configuration is to collect and analyze the ratio of reflected (0 order, for instance) intensity from the phi=0 scan to that of the reflected (0 order) intensity of the phi=90 scan. The intensity ratio is the same as the reflectance ratio, as long as the intensities are measured in quick succession so that there is minimal system drift between the two measurements. In this way the system calibration can be skipped, and the intensity ratio can be analyzed according to the above methods, taking advantage of the reduction in computation expense for the phi=90 case. In some wavelength ranges, the denominator may be close to zero. Those regions need not be analyzed, or the inverse ratio can be analyzed instead. This implementation can be particularly advantageous when using Vacuum Ultra-Violet (VUV) incident light. One implementation of a VUV metrology apparatus is described in U.S. Pat. No. 7,126,131, Broad Band Referencing Reflectometer, by Harrison, the disclosure of which is incorporated herein by reference in its entirety. For such systems, contaminant buildup on calibration standards over time causes difficulties for traditional calibration methods.

An example of a VUV metrology apparatus configured to collect multiple broadband data sets using different azimuthal incident angles is presented in FIG. 4. The instrument is separated into two environmentally controlled chambers, the instrument chamber and the sample chamber. The instrument chamber houses most of the system optics and is not opened to the atmosphere on a regular basis. The sample chamber houses the sample, the sample focusing optic M-2, the reference focusing optic M-4 and the reference plane mirror M-5. This chamber is opened regularly to facilitate changing samples. The instrument is configured to enable collection of sample and reference data sets. The reference data set can be used to correct for system and/or environmental changes which may occur between calibration and sample measurement times. The system may be configured with multiple sources and spectrometers/detectors that are selected using flip-in mirrors FM-1, FM-2, FM-3 and FM-4.

In operation the VUV data is first obtained by switching flip-in source mirrors FM-1 and FM-3 into the “out” position so as to allow light from the VUV source to be collected, collimated and redirected towards beam splitter element BS by focusing mirror M-1. Light striking the beam splitter is divided into two components, the sample beam and the reference beam, using a balanced Michelson interferometer arrangement. The sample beam is reflected from the beam splitter BS and travels through shutter S-1, aperture A-1 and VUV-transparent window W-1. Aperture A-1 is configured to restrict illumination of the sample to some azimuthal plane(s). Shutter S-2 is closed during this time.

Light entering the sample chamber is focused by focusing optic M-2 onto the sample. Light collected from the sample is collimated and redirected by mirror M-2 back through window W-1, aperture A-1 and beam splitter BS. Light passing through the beam splitter encounters aperture A-2, which is configured to selectively pass some fraction of the collected sample response. Light passing through aperture A-2 is redirected and focused onto the entrance slit of the VUV spectrometer by focusing mirror M-3. Flip-in detector mirrors FM-2 and FM-4 are switched to the “out” position during this time.

Following collection of the sample beam, the reference beam is measured by closing shutter S-1 and opening shutter S-2. Once the reference signal has been recorded, data from other spectral regions can be collected in a similar manner using the appropriate flip-in mirrors.

In totality, the use of VUV incident radiation, large polar incident angle or angles, and multiple azimuthal angles, can greatly enhance sensitivity to grating line shape parameters. An analysis using a series of simulations can be done for any given grating structure in order to determine which combination of incident polar angles, azimuthal angles, and wavelengths yields the most information for smallest computation cost.

For faster measurements, where a smaller amount of information may be desired (e.g. line height and average width only), it may be sufficient to use only the phi=90 incidence case and take advantage of the improved calculation speed over the corresponding classical phi=0 incidence case.

Another technique disclosed herein analyzes the multiple sets using different models. For instance phi=90 data might be analyzed using a simpler rectangular line shape model with a course parameter search to narrow down the average parameter values, and then a more thorough analysis done using another one of the spectrum (or all of the spectra together) with a more complicated model to further refine the line shape.

Review of the RCW Method for Conical Incidence

This review follows the notation of the Moharam and Gaylord references cited above. Many publications exist on the basic RCW method, some with notation differences and some with modifications to the formulations/derivation procedures such as that shown in the P. Lalanne and G. M. Morris reference cited above. It should be noted that the phi=90 case reduction described in this disclosure is can be applied to any of these formulations, and is not limited to just that of Moharam and Gaylord.

Note that unreduced eigen-problem matrix and vector indices run from −N to N, with the (−N, −N) matrix element at the top left corner, in order to be consistent with a symmetric diffraction problem with positive and negative orders. When creating a computer algorithm, the indices are labeled from 1 to 2N+1 (or 0 to 2N), depending on the programming language used. It will be recognized, this is a notation preference and has no effect on the outcome. The indices of the reduced matrices will run from 0 to N in either case.

Following the Moharam and Gaylord references, the RCW method expands the fields in each region of FIG. 1 as a generalized Fourier series:

$\begin{matrix} {E_{I} = {E_{inc} + {\sum\limits_{i}{R_{i}{\exp \left\lbrack {{- j}\; \left( {{k_{xi}x} + {k_{y}y} - {k_{I,{zi}}z}} \right)} \right\rbrack}}}}} & {{eq}.\mspace{14mu} 1} \\ {E_{II} = {\sum\limits_{i}{T_{i}\exp \left\{ {- {j\left\lbrack {{k_{xi}x} + {k_{y}y} + {k_{{II},{zi}}\left( {z - d} \right)}} \right\rbrack}} \right\}}}} & {{eq}.\mspace{14mu} 2} \end{matrix}$

in regions I and II, and

$\begin{matrix} {E_{g} = {\sum\limits_{i}{\left\lbrack {{{S_{xi}(z)}x} + {{S_{yi}(z)}y} + {{S_{zi}(z)}z}} \right\rbrack {\exp \left\lbrack {- {j\left( {{k_{xi}x} + {k_{y}y}} \right)}} \right\rbrack}}}} & {{eq}.\mspace{14mu} 3} \\ {H_{g} = {{- {j\left( \frac{ɛ_{f}}{\mu_{f}} \right)}^{1/2}}{\sum\limits_{i}{\left\lbrack {{{U_{xi}(z)}x} + {{U_{yi}(z)}y} + {{U_{zi}(z)}z}} \right\rbrack {\exp \left\lbrack {- {j\left( {{k_{xi}x} + {k_{y}y}} \right)}} \right\rbrack}}}}} & {{eq}.\mspace{14mu} 4} \end{matrix}$

in the grating region, where

$\begin{matrix} {{k_{xi} = {k_{0}\left\lbrack {{n_{I}\sin \; {\theta cos}\; \varphi} - {i\left( {\lambda_{0}/\Lambda} \right)}} \right\rbrack}},} & {{eq}.\mspace{14mu} 5} \\ {{k_{y} = {k_{0}n_{I}\sin \; \theta \; \sin \; \varphi}},} & {{eq}.\mspace{14mu} 6} \\ {k_{I,{zi}} = \left\{ {{{\begin{matrix} \left\lbrack {\left( {k_{0}n_{l}} \right)^{2} - k_{xi}^{2} - k_{y}^{2}} \right\rbrack^{1/2} & {\left( {k_{xi}^{2} + k_{y}^{2}} \right)^{1/2} < {k_{0}n_{l}}} \\ {- {j\left\lbrack {k_{xi}^{2} + k_{y}^{2} - \left( {k_{0}n_{l}} \right)^{2}} \right\rbrack}^{1/2}} & {{\left( {k_{xi}^{2} + k_{y}^{2}} \right)^{1/2} > {k_{0}n_{l}}},} \end{matrix}l} = I},{II},} \right.} & {{eq}.\mspace{14mu} 7} \end{matrix}$

k₀=(2π/λ₀), λ₀ is the incident wavelength, and A is the grating pitch. Note that for a ID grating, k_(y) is constant. In eqs. 3 and 4, ∈_(f) is the permittivity of free space, and μ_(f) is the magnetic permeability of free space.

In equations 1 and 2, the R_(i) and T_(i) are the Fourier coefficients of the electric field in regions I and II, and correspond to the amplitudes of the reflected and transmitted diffraction orders. The diffracted orders can be propagating or evanescent. The corresponding magnetic fields can be obtained from Maxwell's relations

∇×E=−jωμ _(f) H,

∇×H=jω∈ _(f)∈(x)E,  eq. 8

where ω is the angular frequency, and μ is the magnetic permeability. Usually, one assumes μ=μ_(f).

The complex permittivity in the grating region is also expanded as a Fourier series, which is

$\begin{matrix} {{{ɛ(x)} = {\sum\limits_{h}{ɛ_{h}{\exp \left( {j\frac{2\pi \; h}{\Lambda}} \right)}}}},{ɛ_{0} = {{n_{r\; d}^{2}f} + {n_{g\; r}^{2}\left( {1 - f} \right)}}},{ɛ_{h} = {\left( {n_{r\; d}^{2} - n_{g\; r}^{2}} \right)\frac{\sin \left( {\pi \; {hf}} \right)}{\pi \; h}}}} & {{eq}.\mspace{14mu} 9} \end{matrix}$

for the binary grating structure of the first Moharam and Gaylord reference cited above and shown in FIG. 1. In eq. 9, n_(rd) and n_(gr) are the complex indices of refraction for the lines and spaces, respectively.

Eqs. 3, 4, 8, and 9 combine to give a set of coupled equations:

$\begin{matrix} {\begin{bmatrix} {{\partial S_{y}}/{\partial\left( z^{\prime} \right)}} \\ {{\partial S_{x}}/{\partial\left( z^{\prime} \right)}} \\ {{\partial U_{y}}/{\partial\left( z^{\prime} \right)}} \\ {{\partial U_{x}}/{\partial\left( z^{\prime} \right)}} \end{bmatrix} = {\quad{\begin{bmatrix} 0 & 0 & {K_{y}E^{- 1}K_{x}} & {I - {K_{y}E^{- 1}K_{y}}} \\ 0 & 0 & {{K_{x}E^{- 1}K_{x}} - I} & {{- K_{x}}E^{- 1}K_{y}} \\ {K_{x}K_{y}} & {E - K_{y}^{2}} & 0 & 0 \\ {K_{x}^{2} - E} & {{- K_{x}}K_{y}} & 0 & 0 \end{bmatrix} \times \begin{bmatrix} S_{y} \\ S_{x} \\ U_{y} \\ U_{x} \end{bmatrix}}}} & {{eq}.\mspace{14mu} 10} \end{matrix}$

where K_(x) is a diagonal matrix with elements k_(x)/k₀, K_(y) is a diagonal matrix with elements k_(y)/k₀, E is the permittivity matrix (not to be confused with the electric field), with E_(i,j)=∈_((i-j)), and z′=k₀z.

When a truncation order of N is used, eq. 10 is a system of 4(2N+1)×4(2N+1) coupled equations. The authors of the first Moharam and Gaylord reference cited above further reduce eq. 10 to two 2(2N+1)×(2N+1) sets of equations:

$\begin{matrix} {\begin{bmatrix} {{\partial^{2}S_{y}}/{\partial\left( z^{\prime} \right)^{2}}} \\ {{\partial^{2}S_{x}}/{\partial\left( z^{\prime} \right)^{2}}} \end{bmatrix} = {\quad{{\begin{bmatrix} {K_{x}^{2} + {DE}} & {K_{y}\left\lbrack {{E^{- 1}K_{x}E} - K_{x}} \right\rbrack} \\ {K_{x}\left\lbrack {{E^{- 1}K_{y}E} - K_{y}} \right\rbrack} & {K_{y}^{2} + {BE}} \end{bmatrix}\begin{bmatrix} S_{y} \\ S_{x} \end{bmatrix}},{or}}}} & {{eq}.\mspace{14mu} 11} \\ {{\begin{bmatrix} {{\partial^{2}U_{y}}/{\partial\left( z^{\prime} \right)^{2}}} \\ {{\partial^{2}U_{x}}/{\partial\left( z^{\prime} \right)^{2}}} \end{bmatrix} = {\begin{bmatrix} {K_{y}^{2} + {EB}} & {\left\lbrack {K_{x} - {{EK}_{x}E^{- 1}}} \right\rbrack K_{y}} \\ {\left\lbrack {K_{y} - {{EK}_{y}E^{- 1}}} \right\rbrack K_{x}} & {K_{x}^{2} + {ED}} \end{bmatrix}\begin{bmatrix} U_{y} \\ U_{x} \end{bmatrix}}},} & {{eq}.\mspace{14mu} 12} \end{matrix}$

Where B=K_(x)E⁻¹K_(x)−I and D=K_(y)E⁻¹K_(y)−I.

These last equations are reduced still further into two (2N+1)×(2N+1) sets of equations:

[∂² U _(x)/∂(z′)² ]=[K _(y) ² +A][U _(x)]  eq. 13

and

[∂² S _(x)/∂(z′)² ]=[K _(y) ² +BE][S _(x)],  eq. 14

where A=K_(x) ²−E.

Later, Lalanne and Morris (cited above) were able to improve the convergence of the conical case by replacing the matrix E in the third row, second column of eq. 10 with the inverse of the inverse permittivity matrix, Einv, where Einv_(i,j)=(1/∈)_(i,j):

$\begin{matrix} {\begin{bmatrix} {{\partial S_{y}}/{\partial\left( z^{\prime} \right)}} \\ {{\partial S_{x}}/{\partial\left( z^{\prime} \right)}} \\ {{\partial U_{y}}/{\partial\left( z^{\prime} \right)}} \\ {{\partial U_{x}}/{\partial\left( z^{\prime} \right)}} \end{bmatrix} = {\quad{\begin{bmatrix} 0 & 0 & {K_{y}E^{- 1}K_{x}} & {I - {K_{y}E^{- 1}K_{y}}} \\ 0 & 0 & {{K_{x}E^{- 1}K_{x}} - I} & {{- K_{x}}E^{- 1}K_{y}} \\ {K_{x}K_{y}} & {{Einv}^{- 1} - K_{y}^{2}} & 0 & 0 \\ {K_{x}^{2} - E} & {{- K_{x}}K_{y}} & 0 & 0 \end{bmatrix}\begin{bmatrix} S_{y} \\ S_{x} \\ U_{y} \\ U_{x} \end{bmatrix}}}} & {{eq}.\mspace{14mu} 15} \end{matrix}$

which lead to

[∂² U _(x)/∂(z′)² ]=[K _(y) ² +A][U _(x)],  eq. 16

and

[∂² S _(x)/∂(z′)² ]=[K _(y) ² +BEinv ⁻¹ ][S _(x)]  eq. 17

in place of eqs. 13 and 14.

The new formulation eqs. 16 and 17 are advantageous and it may be noted that E and Einv⁻¹ are not the same matrices when they are truncated. The details can be found in references cited above from P. Lalanne and G. M. Morris; G. Granet and B. Guizal; and L. Li.

Equations 16 and 17 are solved by finding the eigenvalues and eigenvectors of the matrices [K_(y) ²+A] and [K_(y) ²+BEinv⁻¹], which leads to

$\begin{matrix} {{U_{xi} = {\sum\limits_{m = 1}^{{2N} + 1}{w_{1,i,m}\left\{ {{{- c_{1,m}^{+}}{\exp \left( {{- k_{0}}q_{1,m}z} \right)}} + {c_{1,m}^{-}{\exp \left\lbrack {k_{0}{q_{1,m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}}},} & {{eq}.\mspace{14mu} 18} \\ {{{S_{xi}(z)} = {\sum\limits_{m = 1}^{{2N} + 1}{w_{2,i,m}\left\{ {{c_{2,m}^{+}{\exp \left( {{- k_{0}}q_{2,m}z} \right)}} + {c_{2,m}^{-}{\exp \left\lbrack {k_{0}{q_{2,m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}}},} & {{eq}.\mspace{14mu} 19} \\ {{{{S_{yi}(z)}{\sum\limits_{m = 1}^{{2N} + 1}{v_{11,i,m}\left\{ {{c_{1,m}^{+}{\exp \left( {{- k_{0}}q_{1,m}z} \right)}} + {c_{1,m}^{-}{\exp \left\lbrack {k_{0}{q_{1,m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}}} + {\sum\limits_{m = 1}^{{2N} + 1}{v_{12,i,m}\left\{ {{c_{2,m}^{+}{\exp \left( {{- k_{0}}q_{2,m}z} \right)}} + {c_{2,m}^{-}{\exp \left\lbrack {k_{0}{q_{2,m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}}},} & {{eq}.\mspace{14mu} 20} \\ {{{U_{yi}(z)} = {{\sum\limits_{m = 1}^{{2N} + 1}{v_{21,i,m}\left\{ {{{- c_{1,m}^{+}}{\exp \left( {{- k_{0}}q_{1,m}z} \right)}} + {c_{1,m}^{-}{\exp \left\lbrack {k_{0}{q_{1,m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}} + {\sum\limits_{m = 1}^{{2N} + 1}{v_{22,i,m}\left\{ {{{- c_{2,m}^{+}}{\exp \left( {{- k_{0}}q_{2,m}z} \right)}} + {c_{2,m}^{-}{\exp \left\lbrack {k_{0}{q_{2,m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}}}},{where}} & {{eq}.\mspace{14mu} 21} \\ {{V_{11} = {A^{- 1}W_{1}Q_{1}}},} & {{eq}.\mspace{14mu} 22} \\ {{V_{12} = {\left( {k_{y}/k_{0}} \right)A^{- 1}K_{x}W_{2}}},} & {{eq}.\mspace{14mu} 23} \\ {{V_{21} = {\left( {k_{y}/k_{0}} \right)B^{- 1}K_{x}E^{- 1}W_{1}}},} & {{eq}.\mspace{14mu} 24} \\ {{V_{22} = {B^{- 1}W_{2}Q_{2}}},} & {{eq}.\mspace{14mu} 25} \end{matrix}$

Q₁ and Q₂ are diagonal matrices with elements q_(1,m) and q_(2,m), which are the square roots of the 2N+1 eigenvalues of the matrices [K_(y) ²+A] and [K_(y) ²+BEinv⁻¹], and W₁ and W₂ are the (2N+1)×(2N+1) matrices formed by the corresponding eigenvectors, with elements w_(1,i,m) and w_(2,i,m). Eqs. 16-25 constitute the eigen-problem portion of the RWC method given in the first the Moharam and Gaylord reference cited above. It is noted that there are other equivalent formulations of the same eigen-problem that will lead to the same final results.

The constants c_(1,m) ⁺, c_(1,m) ⁻, c_(2,m) ⁺, c_(2,m) ⁻ are determined by matching the tangential electric and magnetic field components at the two boundary regions of the grating. The first the Moharam and Gaylord reference cited above uses a boundary formulation where the field components are rotated into the corresponding diffraction plane, φ_(i), for each diffracted order:

sin ψδ_(i0) +R _(s,k)=cos φ_(i) S _(yi)(0)−sin φ_(i) S _(xi)(0),  eq. 26

j[sin ψn ₁ cos θδ_(i0)−(k _(1,zi) /k ₀)R _(s,i)]=−[cos φ_(i) U _(xi)(0)+sin φ_(i) U _(yi)(0)],  eq. 27

cos ψ cos θδ_(i0) −j[k _(1,zi)/(k ₀ n ₁ ²)]R _(p,i)=cos φ_(i) S _(xi)(0)+sin φ_(i) S _(yi)(0)  eq. 28

−jn ₁ cos ψδ_(i0) +R _(p,i)=−[cos φ_(i) U _(yi)(0)−sin φ_(i) U _(xi)(0)],  eq. 29

where

φ_(i)=tan⁻¹(k _(y) /k _(xi)),  eq. 30

R _(x,i)=cos φ_(i) R _(yi)−sin φ_(i) R _(xi),  eq. 31

R _(p,i)=(j/k ₀)[cos φ_(i)(k _(1,zi) ,R _(xi) +k _(xi) R _(zi))+sin φ_(i)(k _(y) R _(zi) +k _(1,zi) R _(yi))],  eq. 32

at the z=0 boundary, and

cos φ_(i) S _(yi)(d)−sin φ_(i) S _(xi)(d)=T _(x,i),  eq. 33

−[cos φ_(i) U _(xi)(d)+sin φ_(i) U _(yi)(d)]=j(k _(1,zi/k0))T _(s,i),  eq. 34

−[cos φ_(i) U _(yi)(d)−sin φ_(i) U _(xi)(d)]=T _(p,i),  eq. 35

cos φ_(i) S _(xi)(d)+sin φ_(i) S _(yi)(d)=j(k _(1,zi) /k ₀ n ₁ ²)T _(p,i),  eq. 36

T _(s,i)=cos φ_(i) T _(yi)−sin φ_(i) T _(xi),  eq. 37

T _(p,i)=(−j/k ₀)[cos φ_(i)(k _(11,zi) T _(xi) −k _(xi) T _(zi))−sin φ_(i)(−k _(11,zi) T _(yi) +k _(y) T _(zi))]  eq. 38

at the z=d boundary. Note that there is one equation for each spatial harmonic retained in the Fourier expansions. R_(x,i) and R_(p,i) are the components of the reflected electric and magnetic field amplitudes normal to the diffraction plane, and T_(x,i) and T_(p,i) are the transmitted amplitudes.

In matrix form, eqs. 26-29 are

$\begin{matrix} {\begin{bmatrix} {\sin \; \psi \; \delta_{i\; 0}} \\ {j\; \sin \; \psi \; n_{I}\; \cos \; \theta \mspace{11mu} \delta_{i\; 0}} \\ {{- j}\; \cos \; \psi \; n_{I}\; \delta_{i\; 0}} \\ {\cos \; \psi \; \cos \; \theta \; \delta_{i\; 0}} \end{bmatrix} = {\begin{bmatrix} I & 0 \\ {- {jY}_{1}} & 0 \\ 0 & I \\ 0 & {- {jZ}_{1}} \end{bmatrix}{\quad{\begin{bmatrix} R_{s} \\ R_{p} \end{bmatrix} = {\begin{bmatrix} V_{ss} & V_{sp} & {V_{ss}X_{1}} & {V_{sp}X_{2}} \\ W_{ss} & W_{sp} & {{- W_{ss}}X_{1}} & {{- W_{sp}}X_{2}} \\ W_{p\; s} & W_{pp} & {{- W_{p\; s}}X_{1}} & {{- W_{pp}}X_{2}} \\ V_{p\; s} & V_{pp} & {V_{\; {p\; s}}X_{1}} & {V_{pp}X_{2}} \end{bmatrix}\begin{bmatrix} c_{1}^{+} \\ c_{2}^{+} \\ c_{1}^{-} \\ c_{2}^{-} \end{bmatrix}}}}}} & {{eq}.\mspace{14mu} 39} \end{matrix}$

for the z=0 boundary and eqs. 33-36 are

$\begin{matrix} {{\begin{bmatrix} {V_{ss}X_{1}} & {V_{sp}X_{2}} & V_{ss} & V_{sp} \\ {W_{ss}X_{1}} & {W_{sp}X_{2}} & {- W_{ss}} & {- W_{sp}} \\ {W_{p\; s}X_{1}} & {W_{pp}X_{2}} & {- W_{p\; s}} & {- W_{pp}} \\ {V_{p\; s}X_{1}} & {V_{pp}X_{2}} & V_{p\; s} & V_{pp} \end{bmatrix}\begin{bmatrix} c_{1}^{+} \\ c_{2}^{+} \\ c_{1}^{-} \\ c_{2}^{-} \end{bmatrix}} = {\begin{bmatrix} I & 0 \\ {jY}_{11} & 0 \\ 0 & I \\ 0 & {jZ}_{11} \end{bmatrix}\begin{bmatrix} T_{s} \\ T_{p} \end{bmatrix}}} & {{eq}.\mspace{14mu} 40} \end{matrix}$

for the z=d boundary, where

V_(ss)=F_(c)V₁₁ W_(pp)=F_(c)V₂₂

W _(ss) =F _(c) W ₁ +F _(s) V ₂₁ V _(pp) =F _(c) W ₂ +F _(s) V ₁₂

V _(sp) =F _(c) V ₁₂ −F _(s) W ₂ W _(ps) =F _(c) V ₂₁ −F _(s) W ₁

W_(sp)=F_(s)V₂₂ V_(ps)=F_(s)V₁₁  eq. 41

Y₁, Y₁₁, Z₁, and Z₁₁ are diagonal matrices with elements (k_(1,zi)/k₀), (k_(1,zi)/k₀), (k_(1,zi)/k₀n₁ ²), and (k_(11,zi)/k₀n₁₁ ²), respectively, X₁ and X₂ are diagonal matrices with elements exp(−k₀q_(1,m)d) and exp(−k₀q_(2,m)d), respectively, and F_(c) and F_(s) are diagonal matrices with elements cos φ_(i) and sin φ_(i), respectively.

Eqs. 39 and 40 are typically solved by eliminating R_(s) and R_(p) from eq. 39, T_(s) and T_(p) from eq. 40 and solving the resulting 4(2N+1) equations for the 4(2N+1) coefficients c_(1,m) ⁺, c_(1,m) ⁻, c_(2,m) ⁺, c_(2,m) ⁻ which can be substituted back into 39 and 40 to solve for the reflected and transmitted amplitudes.

Alternately, a procedure similar to the partial solution approach given in the second Moharam and Gaylord reference cited above can be used to determine reflected amplitudes only, giving a 2(2N+1)×2(2N+1) system of equations for the c_(1,m) ⁺ and c_(2,m) ⁺ coefficients:

$\begin{matrix} {{{{jY}_{1}\mspace{14mu} \sin \; \psi \; \delta_{i\; 0}} + {j\; \sin \; \psi \mspace{11mu} n_{I}\cos \; \theta \; \delta_{i\; 0}}} = {\left\lbrack {{{jY}_{I}f_{T}} + f_{B}} \right\rbrack \begin{bmatrix} c_{1}^{+} \\ c_{2}^{+} \end{bmatrix}}} & {{eq}.\mspace{14mu} 42} \\ {{{\left( Z_{I} \right)_{0,0}\cos \; \psi \; n_{I}\delta_{i\; 0}} + {\cos \; \psi \; \cos \; \theta \; \delta_{i\; 0}}} = {\left\lbrack {{{jZ}_{I}g_{T}} + g_{B}} \right\rbrack \begin{bmatrix} c_{1}^{+} \\ c_{2}^{+} \end{bmatrix}}} & {{eq}.\mspace{14mu} 43} \end{matrix}$

which are related to the reflected amplitudes:

$\begin{matrix} {{R_{s} = {{f_{T}\begin{bmatrix} c_{1}^{+} \\ c_{2}^{+} \end{bmatrix}} - {\sin \; {\psi\delta}_{i\; 0}}}},} & {{eq}.\mspace{14mu} 44} \\ {{R_{p} = {{g_{T}\begin{bmatrix} c_{1}^{+} \\ c_{2}^{+} \end{bmatrix}} + {j\; \cos \; \psi \; n_{I}\delta_{i\; 0}}}},{where}} & {{eq}.\mspace{14mu} 45} \\ {{\begin{bmatrix} f_{T} \\ f_{B} \\ g_{T} \\ g_{B} \end{bmatrix} = {\begin{bmatrix} V_{ss} & V_{sp} \\ W_{ss} & W_{sp} \\ W_{p\; s} & W_{pp} \\ V_{p\; s} & V_{pp} \end{bmatrix} + {\begin{bmatrix} {V_{ss}X_{1}} & {V_{sp}X_{2}} \\ {{- W_{ss}}X_{1}} & {{- W_{sp}}X_{2}} \\ {{- W_{p\; s}}X_{1}} & {{- W_{pp}}X_{2}} \\ {V_{p\; s}X_{1}} & {V_{pp}X_{2}} \end{bmatrix} \cdot a}}},} & {{eq}.\mspace{14mu} 46} \end{matrix}$

and the matrix a is defined as the top half of

$\begin{matrix} {{\begin{bmatrix} {- V_{ss}} & {- V_{sp}} & I & 0 \\ W_{ss} & W_{sp} & {jY}_{11} & 0 \\ W_{p\; s} & W_{pp} & 0 & I \\ {- V_{p\; s}} & V_{pp} & 0 & {jZ}_{11} \end{bmatrix}^{- 1}\begin{bmatrix} {V_{ss}X_{1}} & {V_{sp}X_{2}} \\ {W_{ss}X_{1}} & {W_{sp}X_{2}} \\ {W_{p\; s}X_{1}} & {W_{pp}X_{2}} \\ {V_{p\; s}X_{1}} & {V_{pp}X_{2}} \end{bmatrix}} \equiv {\begin{bmatrix} a \\ b \end{bmatrix}.}} & {{eq}.\mspace{14mu} 47} \end{matrix}$

Note that (Y₁)_(0,0) and (Z₁)_(0,0) refer to the center elements of the matrices Y₁, or (k_(1,zi)/k₀), and Z₁, or (k_(1,zi)/k₀n₁ ²), respectively.

The boundary matching can be generalized to multiple layers using (for example) the enhanced transmittance matrix approach outlined in the second the Moharam and Gaylord reference cited above. Given and L layer stack, where L+1 refers to the substrate, start by setting

$\begin{matrix} {\begin{bmatrix} f_{{L + 1},T} \\ f_{{L + 1},B} \\ g_{{L + 1},T} \\ g_{{L + 1},B} \end{bmatrix} = {\begin{bmatrix} 1 & 0 \\ {j\; Y_{11}} & 0 \\ 0 & 1 \\ 0 & {jZ}_{11} \end{bmatrix}.}} & {{eq}.\mspace{14mu} 48} \end{matrix}$

The matrices for a_(L) and b_(L) are constructed for layer L,

$\begin{matrix} {{{\begin{bmatrix} {- V_{{ss},L}} & {- V_{{sp},L}} & f_{{L + 1},T} \\ W_{{ss},L} & W_{{sp},1} & f_{{L + 1},B} \\ W_{{p\; s},L} & W_{{pp},L} & g_{{L + 1},T} \\ {- V_{{ps},L}} & V_{{pp},L} & g_{{L + 1},B} \end{bmatrix}^{- 1}\begin{bmatrix} {V_{{ss},L}X_{1,L}} & {V_{{sp},L}X_{2,L}} \\ {W_{{ss},L}X_{1,L}} & {W_{{sp},L}X_{2,L}} \\ {W_{{p\; s},L}X_{1,L}} & {W_{{pp},L}X_{2,L}} \\ {V_{{p\; s},L}X_{1,L}} & {V_{{pp},L}X_{2,L}} \end{bmatrix}} \equiv \begin{bmatrix} a_{L} \\ b_{L} \end{bmatrix}},} & {{eq}.\mspace{14mu} 49} \end{matrix}$

where W_(L) and V_(L) come from the solution to the eigen-problem for layer L, X_(1,L)=exp(−k₀q_(1,m,L)d_(L)), and X_(2,L)=exp(−k₀q_(2,m,L)d_(L)), where d_(L) is the thickness of layer L. f_(L) and g_(L) are then obtained from

$\begin{matrix} {\begin{bmatrix} f_{L,T} \\ f_{L,B} \\ g_{L,T} \\ g_{L,B} \end{bmatrix} = {\begin{bmatrix} V_{{ss},L} & V_{{sp},L} \\ W_{{ss},L} & W_{{sp},L} \\ W_{{p\; s},L} & W_{{pp},L} \\ V_{{p\; s},L} & V_{{pp},L} \end{bmatrix} + {\begin{bmatrix} {V_{{ss},L}X_{1,L}} & {V_{{sp},L}X_{2,L}} \\ {{- W_{{ss},L}}X_{1,L}} & {{- W_{{sp},L}}X_{2,L}} \\ {{- W_{{p\; s},L}}X_{1,L}} & {{- W_{{pp},L}}X_{2,L}} \\ {V_{{p\; s},L}X_{1,L}} & {V_{{pp},L}X_{2,L}} \end{bmatrix} \cdot {a_{L}.}}}} & {{eq}.\mspace{14mu} 50} \end{matrix}$

f_(L) and g_(L) are fed back into eq. 49 along with the solution to the eigen-problem for layer L-1 to find a_(L−1) and b_(L−1), and so on until at the top layer f_(1T), f_(1B), g_(1T), and g_(1B) are obtained. These are substituted into eqs. 42-45 in place of f_(T), f_(B), g_(T), and g_(B) to solve for the coefficients c_(1,m) ⁺ and c_(2,m) ⁺ for the top layer, and finally for the reflection coefficients for the diffracted orders via eqs. 44 and 45.

Eqs. 42 and 43 reduce the boundary problem to a 2(2N+1)×2(2N+1) set of equations.

For large truncation order, the boundary problem can still be dominated by the 4(2N+1)×4(2N+1) matrix inversion in eq. 49, but efficient inversion techniques can be employed since only the top half of the matrix is used.

Therefore, for a given incident polar angle theta, the computational expense incurred by using nonzero azimuthal incidence phi is two (2N+1)×(2N+1) eigen-problems versus one (2N+1)×(2N+1) eigen-problem in the corresponding classical (same theta, phi=0) case, a 2(2N+1)×2(2N+1) linear system of equations for the boundary problem (to solve for reflected amplitudes only) versus a (2N+1)×(2N+1) system of equations in the corresponding classical incidence case, and a 4(2N+1)×4(2N+1) matrix inversion in the boundary problem versus a 2(2N+1)×2(2N+1) matrix inversion in the corresponding classical mount. Since these operations are governed by order n³ operations, the conical mount requires approximately 2 times the computing time as the corresponding classical mount case for the eigen-problem, and approximately 8 times the computing time for the boundary problem.

Details of the Reduction in RCW Computation Time for the phi=90 Conical Mount

For the purposes of this description, the symmetry properties of the Fourier series are assumed a priori, and not proved. The initial assumptions can be derived through symmetry arguments, or by experimentation with the conventional formulation given above. In particular, for the phi=90 mount and s polarized incident light (psi=90),

E_(x,i)=E_(x,−i)  eq. 51

E _(y,i) =−E _(y,−i)  eq. 52

H _(x,i) =−H _(x,−i)  eq. 53

H_(y,i)=H_(y,−i)  eq. 54

while for p polarized incident light (psi=0),

E _(x,i) =−E _(x,−i)  =eq. 55

E_(y,i)=E_(y,−i)  eq. 56

H_(x,i)=H_(x,−i)  eq. 57

H _(y,i) =−H _(y,−i)  eq. 58

In equations 51-58, the subscript i refers to the expansion term, which in the incident region corresponds to the diffraction order.

These relationships are valid in all regions of the grating problem, and all of the Fourier expansions can be reduced accordingly. In addition to these relationships, there is a 180 degree phase difference between opposite odd orders, but this can be ignored when not considering interference between multiple gratings.

Also, for phi=90, eq. 5 becomes

k _(xi) =−ik ₀(λ₀/Λ)  eq. 59

This gives

k _(xi) =−k _(x−i)  eq. 60

k_(1,zi)=k_(1,z−i)  eq. 61

The relations 51-61 show that for the phi=90 incidence case:

-   -   i) The generalized Fourier expansions in eqs. 1-4 become regular         Fourier expansions, and     -   ii) The Fourier expansions for the fields have either even or         odd symmetry, depending on the particular field component.

This means that a complex Fourier series representation is not necessary, and the field components can be expressed as cosine series for even symmetry cases or sine series for odd symmetry cases—although the Fourier coefficients themselves will still in general be complex. In either case, the entire content of the 2N+1 terms of a truncated complex Fourier series is contained in the N+1 terms of a cosine or sine series, depending on the symmetry. The usefulness of this for the grating problem arises from the fact the fields have this symmetry in every region. When re-expressed as cosine and sine series, all of the information about the grating problem is contained in roughly half the number of terms required for the traditional formulation. This leads to a reduction in computation time by a factor of approximately 8 compared with the usual phi=90 formulation.

Each incident polarization case will be treated separately. For s polarized light, incident at polar angle theta and the phi=90 conical plane, eqs. 51-54 give

S_(x,i)=S_(x,−i)  eq. 62

S _(y,i) =−S _(y,−i)  eq. 63

U _(x,i) =−U _(x,−i)  eq. 64

U_(y,i)=U_(y,−i)  eq. 65

in the grating region.

The reduced Eqs. 26-29 and eqs. 33-36 may be derived by substituting the symmetry relations for R_(i) and T_(i) into eqs. 1-4 (this requires determining further symmetry relations for R_(z,i) and T_(z,i)), reducing the fields everywhere to the appropriate Fourier cosine or sine series, applying the boundary conditions to the tangential components of the fields at z=0 and z=d, and rotating the boundary equations into the diffraction plane. However, the notation is unnecessarily cumbersome, and it is easier to apply eqs. 62-65 directly to eqs. 26-29 and eqs. 33-36 and use

R_(x,i)=R_(x,−i)  eq. 66

R _(p,i) =−R _(p,−i)  eq. 67

T_(x,i)=T_(x,−i)  eq. 68

T _(p,i) =−T _(p,−i)  eq. 69

to derive the same thing. Eqs. 66-69 can again be verified using the conventional formulation with the phi=90 mount.

In addition, eq. 30 for phi=90 gives

cos φ_(i)=−cos φ⁻¹  eq. 70

and

sin φ_(i)=sin φ_(−i).  eq. 71

Applying the symmetry relations, the i=0 terms in eqs. 26-29 and eqs. 33-36 remain the same, but the nonzero i terms can be combined by adding the i and −i terms of eqs. 26, 27, 33, and 34, and subtracting the −ith from the ith terms in eqs. 28, 29, 35, and 36.

For example, eq. 26 gives

sin ψ+R _(s,0) =−S _(x,0)  (0) eq. 72

for i=0, and

R _(x,i) +R _(s,−i)=cos φ_(i) S _(y,i)(0)+cos φ_(−i) S _(y,−i)(0)−sin φ_(i) S _(x,i)(0)−sin φ_(−i) S _(x,−i)(0),

2R _(s,i)=2 cos φ_(i) S _(y,i)(0)−2 sin φ_(i) S _(x,i)(0),

R _(s,i)=cos φ_(i) S _(y,i)(0)−sin φ_(i) S _(x,i)(0),  eq. 73

which is the same as eq. 26, except that i>0.

Similarly, eq. 28 gives

$\begin{matrix} {{{{\cos \; {\psi cos}\; \theta} - {{j\left\lbrack {k_{I,{z\; 0}}/\left( {k_{0}n_{I}^{2}} \right)} \right\rbrack}R_{p,0}}} = {S_{y,0}(0)}}{{{for}\mspace{14mu} i} = 0},{{{and} - {{j\left\lbrack {k_{I,{zi}}/\left( {k_{0}n_{I}^{2}} \right)} \right\rbrack}R_{p,i}} + {{j\left\lbrack {k_{I,{z{({- i})}}}/\left( {k_{0}n_{I}^{2}} \right)} \right\rbrack}R_{p,{- i}}}} = {{\cos \; \varphi_{i}{S_{x,i}(0)}} - {\cos \; \varphi_{- i}{S_{x,{- i}}(0)}} + {\sin \; \varphi_{i}{S_{y,i}(0)}} - {\sin \; \varphi_{- i}{S_{y,{- i}}(0)}}}},} & {{eq}.\mspace{14mu} 74} \\ \begin{matrix} {{{{- 2}{j\left\lbrack {k_{I,{zi}}/\left( {k_{0}n_{I}^{2}} \right)} \right\rbrack}R_{p,i}} = {{2\; \cos \; \varphi_{i}{S_{x,i}(0)}} + {2\sin \; \varphi_{i}{S_{y,i}(0)}}}}, -} \\ {{{j\left\lbrack {k_{I,{zi}}/\left( {k_{0}n_{I}^{2}} \right)} \right\rbrack}R_{p,i}}} \\ {{= {{\cos \; \varphi_{i}{S_{x,i}(0)}} + {\sin \; \varphi_{i}S_{y,i}(0)}}},} \end{matrix} & {{eq}.\mspace{14mu} 75} \end{matrix}$

which is eq. 28, but with i>0.

The other boundary equations can be similarly reduced, and the form of the boundary problem is the same as the conventional one, except that only the i=0 and i>0 terms occur in the matrix equations. This reduces the matrix boundary problem (eqs. 39 and 40) to 4(N+1)×4(N+1) systems of equations, but leaves the form the same as in eqs. 26-41, as long as it is possible to also reduce the solution in the grating region to determining 4(N+1) coefficients c_(1,m) ⁺, c_(1,m) ⁻, c_(2,m) ⁺, c_(2,m) ⁻ instead of 4(2N+1) coefficients. This is shown to be the case below. Therefore, except for modifying the matrices to consist of N+1 harmonic terms (and therefore N+1 diffraction orders), the boundary problem is the same as previously defined. Now, R_(xi), R_(pi), T_(xi), and T_(pi) are the amplitudes of both the +i and −i diffracted orders.

To reduce the eigen-system, apply eqs. 60 and 62-65 directly to equations 15-17, reducing the total number of unknowns from 4(2N+1) to 4(N+1). The eigen-problems specified by eqs. 16 and 17 are each reduced from size (2N+1)×(2N+1) to size (N+1)×(N+1), for a total reduction of a factor of approximately 8 over the previous conical descriptions, and a factor of 4 over the corresponding classical mount eigen-problem.

Aside from reducing eqs. 16 and 17, reduced matrices for eqs. 22-25 will also need to be found. This will reduce the solution in the grating region to the determination of 4(N+1) coefficients instead of 4(N+1).

Note that the symmetry of the lamellar grating also implies that the elements of the permittivity matrix satisfy

E_(i,j)=E_(−i,−j).  eq. 76

The i th row of equation 16 can be written as

$\begin{matrix} {\frac{\partial^{2}U_{xi}}{\partial\left( z^{\prime} \right)^{2}} = {{\frac{k_{y}^{2}}{k_{0}}U_{xi}} + {\frac{k_{xi}^{2}}{k_{0}^{2}}U_{xi}} - {\sum\limits_{m = {- \infty}}^{\infty}{E_{i,m}{U_{xm}.}}}}} & {{eq}.\mspace{14mu} 77} \end{matrix}$

Due to eq. 64, eq. 16 obeys the following symmetry condition:

$\begin{matrix} {\frac{\partial^{2}U_{xi}}{\partial\left( z^{\prime} \right)^{2}} = {- {\frac{\partial^{2}U_{x - i}}{\partial\left( z^{\prime} \right)^{2}}.}}} & {{eq}.\mspace{14mu} 78} \end{matrix}$

Subtracting the −i th row from the i th row gives

$\begin{matrix} {{\frac{\partial^{2}U_{x\; 0}}{\partial\left( z^{\prime} \right)^{2}} = {{\frac{k_{y}^{2}}{k_{0}^{2}}U_{x\; 0}} - {\sum\limits_{m = {- \infty}}^{\infty}{E_{0,m}U_{xm}}}}}\begin{matrix} {\frac{\partial^{2}U_{x\; 0}}{\partial\left( z^{\prime} \right)^{2}} = {{\frac{k_{y}^{2}}{k_{0}^{2}}U_{x\; 0}} - {E_{0,0}U_{x\; 0}} - {\sum\limits_{m = {- \infty}}^{- 1}{E_{0,m}U_{xm}}} - {\sum\limits_{m = 1}^{\infty}{E_{0,m}U_{xm}}}}} \\ {{= {{\frac{k_{y}^{2}}{k_{0}^{2}}U_{x\; 0}} - {E_{0,0}U_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{E_{0,{- m}}U_{xm}}} - {\sum\limits_{m = 1}^{\infty}{E_{0,m}U_{xm}}}}},} \end{matrix}{so}} & {{eq}.\mspace{14mu} 79} \\ {{\frac{\partial^{2}U_{x\; 0}}{\partial\left( z^{\prime} \right)^{2}} = {{\frac{k_{y}^{2}}{k_{0}^{2}}U_{x\; 0}} - \left\{ {{E_{0,0}U_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\left( {E_{0,m} - E_{0,{- m}}} \right)U_{xm}}}} \right\}}}{{{{for}\mspace{14mu} i} = 0},{and}}\begin{matrix} {{2\frac{\partial^{2}U_{xi}}{\partial\left( z^{\prime} \right)^{2}}} = {{2\frac{k_{y}^{2}}{k_{0}^{2}}U_{xi}} + {2\frac{k_{xi}^{2}}{k_{0}^{2}}U_{xi}} - {\sum\limits_{m = {- \infty}}^{\infty}{E_{i,m}U_{xm}}} +}} \\ {{\sum\limits_{m = {- \infty}}^{\infty}{E_{{- i},m}U_{xm}}}} \\ {= {{2\frac{k_{y}^{2}}{k_{0}^{2}}U_{xi}} + {2\frac{k_{xi}^{2}}{k_{0}^{2}}U_{xi}} - {E_{i,0}U_{x\; 0}} - {\sum\limits_{m = {- \infty}}^{- 1}{E_{i,m}U_{xm}}} -}} \\ {{{\sum\limits_{m = 1}^{\infty}{E_{i,m}U_{xm}}} + {E_{{- i},0}U_{x\; 0}} + {\sum\limits_{m = {- \infty}}^{- 1}{E_{{- i},m}U_{xm}}} +}} \\ {{\sum\limits_{m = 1}^{\infty}{E_{{- i},m}U_{xm}}}} \\ {= {{2\frac{k_{y}^{2}}{k_{0}^{2}}U_{xi}} + {2\frac{k_{xi}^{2}}{k_{0}^{2}}U_{xi}} - {E_{i,0}U_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{E_{i,{- m}}U_{xm}}} -}} \\ {{{\sum\limits_{m = 1}^{\infty}{E_{i,m}U_{xm}}} + {E_{{- i},0}U_{x\; 0}} - {\sum\limits_{m = 1}^{\infty}{E_{{- i},{- m}}U_{xm}}} +}} \\ {{{\sum\limits_{m = 1}^{\infty}{E_{{- i},m}U_{xm}}} + {\sum\limits_{m = 1}^{\infty}{E_{{- i},m}U_{xm}}}}} \\ {= {{2\frac{k_{y}^{2}}{k_{0}^{2}}U_{xi}} + {2\frac{k_{xi}^{2}}{k_{0}^{2}}U_{xi}} - {E_{i,0}U_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{E_{i,{- m}}U_{xm}}} -}} \\ {{{\sum\limits_{m = 1}^{\infty}{E_{i,m}U_{xm}}} + {E_{{- i},0}U_{x\; 0}} - {\sum\limits_{m = 1}^{\infty}{E_{{- i},{- m}}U_{xm}}} +}} \\ {{\sum\limits_{m = 1}^{\infty}{E_{{- i},m}U_{xm}}}} \\ {= {{2\frac{k_{y}^{2}}{k_{0}^{2}}U_{xi}} + {2\frac{k_{xi}^{2}}{k_{0}^{2}}U_{xi}} - {\left( {E_{i,0} - E_{{- i},0}} \right)U_{x\; 0}} -}} \\ {{{\sum\limits_{m = 1}^{\infty}{\left( {E_{i,m} + E_{{- i},{- m}} - E_{i,{- m}} - E_{{- i},m}} \right)U_{xm}}},}} \end{matrix}{so}} & \; \\ {\frac{\partial^{2}U_{xi}}{\partial\left( z^{\prime} \right)^{2}} = {{\frac{k_{y}^{2}}{k_{0}^{2}}U_{xi}} + {\frac{k_{xi}^{2}}{k_{0}^{2}}U_{xi}} - \left\{ {{\frac{1}{2}\left( {E_{i,0} - E_{{- i},0}} \right)U_{x\; 0}} + {\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\left( {E_{i,m} + E_{{- i},{- m}} - E_{{- i},m} - E_{i,{- m}}} \right)U_{xm}}}}} \right\}}} & {{eq}.\mspace{14mu} 80} \end{matrix}$

for i>0. Note that i now runs from 0 to ∞ instead of −∞ to ∞.

The first two terms in eq. 79 and 80 indicate that the matrices K_(y) ² and K_(x) ² in eq. 16 should simply be replaced by diagonal matrices consisting of the 0 and positive terms of the original matrices. In fact, this will turn out to be the case for K_(x) and K_(y) throughout, and the subscripts and superscripts on these matrices distinguishing reduced from unreduced will hereafter be omitted.

The terms

$\begin{matrix} {{E_{i,0}U_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\left( {E_{0,m} - E_{0,{- m}}} \right)U_{xm}}}} & {{eq}.\mspace{14mu} 81} \end{matrix}$

from eq. 79 and

$\begin{matrix} {{\frac{1}{2}\left( {E_{i,0} - E_{{- i},0}} \right)U_{x\; 0}} + {\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\left( {E_{i,m} + E_{{- i},{- m}} - E_{{- i},m} - E_{i,{- m}}} \right)U_{xm}}}}} & {{eq}.\mspace{14mu} 82} \end{matrix}$

from eq. 80 are the rows of the reduced matrix that replaces the matrix E in eq. 16:

$\begin{matrix} {E_{reduced}^{s} = \begin{bmatrix} E_{0,0} & {E_{0,1} - E_{0,{- 1}}} & {E_{0,2} - E_{0,{- 2}}} & \ldots \\ {\frac{1}{2}\left( {E_{1,0} -} \right.} & {\frac{1}{2}\left( {E_{1,1} + E_{{- 1},{- 1}} -} \right.} & {\frac{1}{2}\left( {E_{1,2} + E_{{- 1},{- 2}} -} \right.} & \ldots \\ \left. E_{{- 1},0} \right) & \left. {E_{{- 1},1} - E_{1,{- 1}}} \right) & \left. {E_{{- 1},2} - E_{1,{- 2}}} \right) & \; \\ {\frac{1}{2}\left( {E_{2,0} -} \right.} & {\frac{1}{2}\left( {E_{2,1} + E_{{- 2},{- 1}} -} \right.} & {\frac{1}{2}\left( {E_{2,2} + E_{{- 2},{- 2}} -} \right.} & \ldots \\ \left. E_{{- 2},0} \right) & \left. {E_{{- 2},1} - E_{2,{- 1}}} \right) & \left. {E_{{- 2},2} - E_{1,{- 2}}} \right) & \; \\ \vdots & \; & \; & ⋰ \end{bmatrix}} & {{eq}.\mspace{14mu} 83} \\ {with} & \; \\ {A_{reduced}^{s} = {K_{x}^{2} - E_{reduced}^{s}}} & {{eq}.\mspace{14mu} 84} \\ {and} & \; \\ {{\left\lbrack \frac{\partial^{2}U_{x}}{\partial\left( z^{\prime} \right)^{2}} \right\rbrack = {\left\lbrack {K_{y}^{2} + A_{reduced}^{s}} \right\rbrack \left\lbrack U_{x} \right\rbrack}},} & {{eq}.\mspace{14mu} 85} \end{matrix}$

where the subscript s refers to the incident polarization case. All of the vectors in eq. 85 are of size N+1, and the matrices are of size (N+1)×(N+1) for a given truncation order, N.

Many of the terms in eq. 83 can be reduced using eq. 76, but it is more useful to assume nothing about the elements of the matrices being reduced. This way, other matrices that may not necessarily obey eq. 76 can be reduced using the same formulas. Along these lines, more general reductions can be formulated, which can be applied to a variety of matrices or even the products of matrices that will be required to find the reduced matrices of eqs. 22-25.

Disregarding the simpler diagonal matrices K_(y) and K_(x), the unreduced equations have the general form

$\begin{matrix} {{l\left\lbrack P_{i} \right\rbrack} = {\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{i,m}Q_{m}}}} & {{eq}.\mspace{14mu} 86} \end{matrix}$

Where

is a linear operator, such as

${\frac{\partial}{\partial\left( z^{\prime} \right)}\mspace{14mu} {or}\mspace{14mu} \frac{\partial^{2}}{\partial\left( z^{\prime} \right)^{2}}},$

and the elements of the vectors P and Q are spatial harmonic coefficients of the Fourier expansions for the corresponding fields. The goal is to find a reduced matrix for ∈ through application of symmetry relations to the vectors P and Q. Without making any assumptions about the elements of the matrix ∈, there are in general four types of reductions:

-   -   1) Both P and Q are even and the corresponding Fourier series         can be reduced to cosine series,     -   2) Both P and Q are odd and the corresponding Fourier         expressions can be reduced to sine series,     -   3) P is even and Q is odd,     -   4) P is odd and Q is even.

Note that if P has even or odd symmetry, then

$\frac{\partial P}{\partial\left( z^{\prime} \right)}\mspace{14mu} {and}\mspace{14mu} \frac{\partial^{2}P}{\partial\left( z^{\prime} \right)^{2}}$

are also even or odd, respectively.

The reduction leading to eqs. 81 and 82 belongs to category 2. The same argument can be applied to eq. 86 to give

$\begin{matrix} {{{l\left\lbrack P_{0} \right\rbrack} = {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{0,m} - ɛ_{0,{- m}}} \right)Q_{m}}}}},\mspace{14mu} {i = 0},{and}} & {{eq}.\mspace{14mu} 87} \\ {{{l\left\lbrack P_{i} \right\rbrack} = {{\frac{1}{2}\left( {ɛ_{i,0} - ɛ_{{- i},0}} \right)Q_{0}} + {\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{i,m} + ɛ_{{- i},{- m}} - ɛ_{{- i},m} - ɛ_{i,{- m}}} \right)Q_{m}}}}}},{i > 0},} & {{eq}.\mspace{14mu} 88} \end{matrix}$

for any matrix ∈ and field harmonics P and Q having odd symmetry.

The other 3 cases are developed below.

For case 1, both P and Q have even symmetry. Therefore the i and −i rows can be added together:

$\begin{matrix} \begin{matrix} {{l\left\lbrack P_{0} \right\rbrack} = {\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{0,m}Q_{m}}}} \\ {= {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{0,m}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,m}Q_{m}}}}} \\ {= {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,{- m}}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,m}Q_{m}}}}} \end{matrix} & {{eq}.\mspace{14mu} 89} \\ {{{l\left\lbrack P_{0} \right\rbrack} = {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{0,m} + ɛ_{0,{- m}}} \right)Q_{m}}}}},{i = 0},{and}} & \; \\ {\begin{matrix} {{{l\left\lbrack P_{i} \right\rbrack} + {l\left\lbrack P_{- i} \right\rbrack}} = {{\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{i,m}Q_{m}}} + {\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{{- i},m}Q_{m}}}}} \\ {= {{ɛ_{i,0}Q_{0}} + {ɛ_{{- i},0}Q_{0}} + {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{i,m}Q_{m}}} +}} \\ {{{\sum\limits_{m = 1}^{\infty}{ɛ_{i,m}Q_{m}}} + {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{{- i},m}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{{- i},m}Q_{m}}}}} \\ {= {{\left( {ɛ_{i,0} + ɛ_{{- i},0}} \right)Q_{0}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{i,{- m}}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{i,m}Q_{m}}} +}} \\ {{{\sum\limits_{m = 1}^{\infty}{ɛ_{{- i},{- m}}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{{- i},m}Q_{m}}}}} \\ {= {{\left( {ɛ_{i,0} + ɛ_{{- i},0}} \right)Q_{0}} + {\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{i,m} + ɛ_{i,{- m}} + ɛ_{{- i},m} + ɛ_{{- i},{- m}}} \right)Q_{m}}}}} \\ {{= {2\; {l\left\lbrack P_{i} \right\rbrack}}},{so}} \end{matrix}} & \; \\ \begin{matrix} {{l\left\lbrack P_{0} \right\rbrack} = {\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{0,m}Q_{m}}}} \\ {= {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{0,m}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,m}Q_{m}}}}} \\ {= {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,{- m}}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,m}Q_{m}}}}} \end{matrix} & {{eq}.\mspace{14mu} 89} \\ \square & \square \\ \square & \square \\ \square & \square \end{matrix}$

Eqs. 89 and 90 define a reduced matrix

$\begin{matrix} {ɛ_{reduced} = \begin{bmatrix} ɛ_{0,0} & {ɛ_{0,1} + ɛ_{0,{- 1}}} & {ɛ_{0,2} + ɛ_{0,{- 2}}} & \ldots \\ {\frac{1}{2}\left( {ɛ_{1,0} +} \right.} & {\frac{1}{2}\left( {ɛ_{1,1} + ɛ_{{- 1},{- 1}} +} \right.} & {\frac{1}{2}\left( {ɛ_{1,2} + ɛ_{{- 1},{- 2}} +} \right.} & \ldots \\ \left. ɛ_{{- 1},0} \right) & \left. {ɛ_{1,{- 1}} + ɛ_{{- 1},1}} \right) & \left. {ɛ_{1,{- 2}} + ɛ_{{- 1},2}} \right) & \; \\ {\frac{1}{2}\left( {ɛ_{2,0} +} \right.} & {\frac{1}{2}\left( {ɛ_{2,1} + ɛ_{{- 2},{- 1}} +} \right.} & {\frac{1}{2}\left( {ɛ_{2,2} + ɛ_{{- 2},{- 2}} +} \right.} & \ldots \\ \left. ɛ_{{- 2},0} \right) & \left. {ɛ_{2,{- 1}} + ɛ_{{- 2},1}} \right) & {\left. {ɛ_{2,{- 2}} + ɛ_{{- 1},2}} \right)\;} & \; \\ \vdots & \; & \; & ⋰ \end{bmatrix}} & {{eq}.\mspace{14mu} 91} \end{matrix}$

For case 3, the i and −i rows are again added:

$\begin{matrix} \begin{matrix} {{l\left\lbrack P_{0} \right\rbrack} = {\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{0,m}Q_{m}}}} \\ {= {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{0,m}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,m}Q_{m}}}}} \\ {{= {{ɛ_{0,0}Q_{0}} - {\sum\limits_{m = 1}^{\infty}{ɛ_{0,{- m}}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,m}Q_{m}}}}},} \end{matrix} & {{eq}.\mspace{14mu} 92} \\ {{{l\left\lbrack P_{0} \right\rbrack} = {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{0,m} - ɛ_{0,{- m}}} \right)Q_{m}}}}},} & \; \\ {{i = 0},} & \; \\ {and} & \; \\ \begin{matrix} {{{l\left\lbrack P_{i} \right\rbrack} + {l\left\lbrack P_{- i} \right\rbrack}} = {{\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{i,m}Q_{m}}} + {\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{{- i},m}Q_{m}}}}} \\ {= {{ɛ_{i,0}Q_{0}} + {ɛ_{{- i},0}Q_{0}} + {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{i,m}Q_{m}}} +}} \\ {{{\sum\limits_{m = 1}^{\infty}{ɛ_{i,m}Q_{m}}} + {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{{- i},m}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{{- i},m}Q_{m}}}}} \\ {= {{\left( {ɛ_{i,0} + ɛ_{{- i},0}} \right)Q_{0}} - {\sum\limits_{m = 1}^{\infty}{ɛ_{i,{- m}}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{i,m}Q_{m}}} -}} \\ {{{\sum\limits_{m = 1}^{\infty}{ɛ_{{- i},{- m}}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{{- i},m}Q_{m}}}}} \\ {= {{\left( {ɛ_{i,0} + ɛ_{{- i},0}} \right)Q_{0}} + {\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{i,m} + ɛ_{{- i},m} - ɛ_{i,{- m}} - ɛ_{{- i},{- m}}} \right)Q_{m}}}}} \\ {{= {2\; {l\left\lbrack P_{i} \right\rbrack}}},} \end{matrix} & \; \\ {giving} & \; \\ {{{l\left\lbrack P_{i} \right\rbrack} = {{\frac{1}{2}\left( {ɛ_{i,0} + ɛ_{{- i},0}} \right)Q_{0}} + {\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{i,m} + ɛ_{{- i},m} - ɛ_{i,{- m}} - ɛ_{{- i},{- m}}} \right)Q_{m}}}}}},} & {{eq}.\mspace{14mu} 93} \\ {i > 0.} & \; \end{matrix}$

For case 4, subtract the −i th row from the i th row:

$\begin{matrix} \begin{matrix} {{l\left\lbrack P_{0} \right\rbrack} = {\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{0,m}Q_{m}}}} \\ {= {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{0,m}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,m}Q_{m}}}}} \\ {{= {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,{- m}}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{0,m}Q_{m}}}}},} \end{matrix} & {{eq}.\mspace{14mu} 94} \\ {{{l\left\lbrack P_{0} \right\rbrack} = {{ɛ_{0,0}Q_{0}} + {\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{0,m} + ɛ_{0,{- m}}} \right)\; Q_{m}}}}},\mspace{11mu} {i = 0},} & \; \\ {and} & \; \\ \begin{matrix} {{{l\left\lbrack P_{i} \right\rbrack} - {l\left\lbrack P_{- i} \right\rbrack}} = {{\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{i,m}Q_{m}}} - {\sum\limits_{m = {- \infty}}^{\infty}{ɛ_{{- i},m}Q_{m}}}}} \\ {= {{ɛ_{i,0}Q_{0}} - {ɛ_{{- i},0}Q_{0}} + {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{i,m}Q_{m}}} +}} \\ {{{\sum\limits_{m = 1}^{\infty}{ɛ_{i,m}Q_{m}}} - {\sum\limits_{m = {- \infty}}^{- 1}{ɛ_{{- i},m}Q_{m}}} - {\sum\limits_{m = 1}^{\infty}{ɛ_{{- i},m}Q_{m}}}}} \\ {= {{\left( {ɛ_{i,0} - ɛ_{{- i},0}} \right)Q_{0}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{i,{- m}}Q_{m}}} + {\sum\limits_{m = 1}^{\infty}{ɛ_{i,m}Q_{m}}} -}} \\ {{{\sum\limits_{m = 1}^{\infty}{ɛ_{{- i},{- m}}Q_{m}}} - {\sum\limits_{m = 1}^{\infty}{ɛ_{{- i},m}Q_{m}}}}} \\ {= {{\left( {ɛ_{i,0} - ɛ_{{- i},0}} \right)Q_{0}} +}} \\ {{\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{i,m} + ɛ_{i,{- m}} - ɛ_{{- i},m} - ɛ_{{- i},{- m}}} \right)Q_{m}}}} \\ {{= {2\; {l\left\lbrack P_{i} \right\rbrack}}},} \end{matrix} & \; \\ {giving} & \; \\ \begin{matrix} {{l\left\lbrack P_{i} \right\rbrack} = {{\frac{1}{2}\left( {ɛ_{i,0} - ɛ_{{- i},0}} \right)Q_{0}} +}} \\ {{{\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\left( {ɛ_{i,m} + ɛ_{i,{- m}} - ɛ_{{- i},m} - ɛ_{{- i},{- m}}} \right)Q_{m}}}},\mspace{14mu} {i > 0.}}} \end{matrix} & {{eq}.\mspace{14mu} 95} \end{matrix}$

Application of case 2 with ∈=E leads directly to eqs. 81 and 82 for E_(reduced) and leads to the reduced eigenproblem of eqs. 84 and 85. To reduce eq. 17, case 1 can be applied directly to the product BEinv⁻¹, giving

$\begin{matrix} {\frac{\partial^{2}S_{x\; 0}}{\partial\left( z^{\prime} \right)^{2}} = {{\left( \frac{k_{y}^{2}}{k_{0}} \right)S_{x\; 0}} + {\left( {BEinv}^{- 1} \right)_{0,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\left\lbrack {\left( {BEinv}^{- 1} \right)_{0,m} + \left( {BEinv}^{- 1} \right)_{0,{- m}}} \right\rbrack S_{xm}}}}} & {{eq}.\mspace{14mu} 96} \\ {\frac{\partial^{2}S_{xi}}{\partial\left( z^{\prime} \right)^{2}} = {{\left( \frac{k_{y}^{2}}{k_{0}} \right)S_{x,i}} + {{\frac{1}{2}\left\lbrack {\left( {BEinv}^{- 1} \right)_{i,0} + \left( {BEinv}^{- 1} \right)_{{- i},0}} \right\rbrack}S_{x\; 0}} + {\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\left\lbrack {\left( {BEinv}^{- 1} \right)_{i,m} + \left( {BEinv}^{- 1} \right)_{i,{- m}} + \left( {BEinv}^{- 1} \right)_{{- i},m} + \left( {BEinv}^{- 1} \right)_{{- i},{- m}}} \right\rbrack {S_{xm}.}}}}}} & {{eq}.\mspace{14mu} 97} \end{matrix}$

This involves (2N+1)×(2N+1) matrix multiplications to find the elements of BE_(inv) ⁻¹. A slightly more efficient way to construct the reduced eigen-problem is to reduce the components of the product first, and multiply the reduced (N+1)×(N+1) matrices together to form B_(reduced)(Einv⁻¹)_(reduced).

To do this one can go back to eq. 15 and apply the appropriate reductions to the third column of the second row and second column of the third row for B and Einv⁻¹, respectively.

For B, explicitly reduce the product K_(x)E⁻¹K_(x):

$\begin{matrix} {{\frac{\partial S_{xi}}{\partial\left( z^{\prime} \right)} = {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}}}},} & {{eq}.\mspace{14mu} 98} \end{matrix}$

where the dots replace other terms in eq. 15 that are not relevant for the purpose of finding the reduced matrix.

Adding the i th and −i th rows:

$\begin{matrix} {{\frac{\partial S_{x\; 0}}{\partial\left( z^{\prime} \right)} = {\ldots + 0}},{i = 0},} & {{eq}.\mspace{14mu} 99} \\ {{{{since}\mspace{14mu} k_{x\; 0}} = 0},{and}} & \; \\ \begin{matrix} {{2\frac{\partial S_{xi}}{\partial\left( z^{\prime} \right)}} = {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} +}} \\ {{\frac{k_{x - i}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}\left\lbrack {{\sum\limits_{m = {- \infty}}^{- 1}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack} +}} \\ {{\frac{k_{x - i}}{k_{0}}\left\lbrack {{\sum\limits_{m = {- \infty}}^{- 1}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}\left\lbrack {{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,{- m}}\frac{k_{x - m}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack} +}} \\ {{\frac{k_{x - i}}{k_{0}}\left\lbrack {{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},{- m}}\frac{k_{x - m}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}\left\lbrack {{- {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack} -}} \\ {{\frac{k_{xi}}{k_{0}}\left\lbrack {{- {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}\left\lbrack {{- {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}} +} \right.}}} \\ \left. {{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}} - {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = 1}^{\infty}\left\lbrack {\left( E^{- 1} \right)_{i,m} + \left( E^{- 1} \right)_{{- i},{- m}} - \left( E^{- 1} \right)_{i,{- m}} -} \right.}}}} \\ {\left. \left( E^{- 1} \right)_{{- i},m} \right\rbrack \frac{k_{xm}}{k_{0}}U_{ym}} \\ {{= {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = 1}^{\infty}{\left\lbrack {{2\left( E^{- 1} \right)_{i,m}} - {2\left( E^{- 1} \right)_{i,{- m}}}} \right\rbrack \frac{k_{xm}}{k_{0}}U_{ym}}}}}},} \end{matrix} & \; \\ {giving} & \; \\ {\frac{\partial S_{xi}}{\partial\left( z^{\prime} \right)} = {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = 1}^{\infty}\left\lbrack {{\left( E^{- 1} \right)_{i,m} - {\left( E^{- 1} \right)_{i,{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}},} \right.}}}} & {{eq}.\mspace{14mu} 100} \\ {{i > 0},} & \; \\ {or} & \; \\ {\left( {K_{x}E^{- 1}K_{x}} \right)_{reduced}^{s} = \begin{bmatrix} 0 & 0 & 0 & \cdots \\ 0 & \begin{matrix} {\frac{k_{x\; 1}}{k_{0}}\left\lbrack {\left( E^{- 1} \right)_{1,1} -} \right.} \\ {\left. \left( E^{- 1} \right)_{1,{- 1}} \right\rbrack \frac{k_{x\; 1}}{k_{0}}} \end{matrix} & \begin{matrix} {\frac{k_{x\; 1}}{k_{0}}\left\lbrack {\left( E^{- 1} \right)_{1,2} -} \right.} \\ {\left. \left( E^{- 1} \right)_{1,{- 2}} \right\rbrack \frac{k_{x\; 2}}{k_{0}}} \end{matrix} & \cdots \\ 0 & \begin{matrix} {\frac{k_{x\; 2}}{k_{0}}\left\lbrack {\left( E^{- 1} \right)_{2,1} -} \right.} \\ {\left. \left( E^{- 1} \right)_{2,{- 1}} \right\rbrack \frac{k_{x\; 1}}{k_{0}}} \end{matrix} & \begin{matrix} {\frac{k_{x\; 2}}{k_{0}}\left\lbrack {\left( E^{- 1} \right)_{2,2} -} \right.} \\ {\left. \left( E^{- 1} \right)_{2,{- 2}} \right\rbrack \frac{k_{x\; 2}}{k_{0}}} \end{matrix} & \cdots \\ \vdots & \; & \; & ⋰ \end{bmatrix}} & {{eq}.\mspace{14mu} 101} \end{matrix}$

in explicit form. Then

B _(reduced) ^(s)=(K _(x) E ⁻¹ K _(x))_(reduced) ^(s) −I.  eq. 102

For Einv⁻¹, use

$\begin{matrix} {\frac{\partial U_{yi}}{\partial z^{\prime}} = {\ldots + {\sum\limits_{m = {- \infty}}^{\infty}{\left( {Einv}^{- 1} \right)_{im}S_{xm}}}}} & {{eq}.\mspace{14mu} 103} \end{matrix}$

to which case 1 may be directly applied:

$\begin{matrix} {\mspace{236mu} {{\frac{\partial U_{y\; 0}}{\partial z^{\prime}} = {\ldots + {\left( {Einv}^{- 1} \right)_{0,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{2\left( {Einv}^{- 1} \right)_{0,m}S_{xm}}}}},{i = 0},}} & {{eq}.\mspace{14mu} 104} \\ {\mspace{236mu} {{\frac{\partial U_{y\; i}}{\partial z^{\prime}} = {\ldots + {\left( {Einv}^{- 1} \right)_{i,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\left\lbrack {\left( {Einv}^{- 1} \right)_{i,m} + \left( {Einv}^{- 1} \right)_{i,{- m}}} \right\rbrack S_{xm}}}}},{i > 0},{{which}\mspace{14mu} {implies}}}} & {{eq}.\mspace{14mu} 105} \\ {{\left( {Einv}^{- 1} \right)_{reduced}^{s} = \begin{bmatrix} \left( {Einv}^{- 1} \right)_{0,0} & {2\left( {Einv}^{- 1} \right)_{0,1}} & {2\left( {Einv}^{- 1} \right)_{0,2}} & \ldots \\ \left( {Einv}^{- 1} \right)_{1,0} & {\left( {Einv}^{- 1} \right)_{1,1} + \left( {Einv}^{- 1} \right)_{1,{- 1}}} & {\left( {Einv}^{- 1} \right)_{1,2} + \left( {Einv}^{- 1} \right)_{1,{- 2}}} & \ldots \\ \left( {Einv}^{- 1} \right)_{2,0} & {\left( {Einv}^{- 1} \right)_{2,1} + \left( {Einv}^{- 1} \right)_{2,{- 1}}} & {\left( {Einv}^{- 1} \right)_{2,2} + \left( {Einv}^{- 1} \right)_{2,{- 2}}} & \ldots \\ \vdots & \; & \; & ⋰ \end{bmatrix}}} & {{eq}.\mspace{14mu} 106} \end{matrix}$

where one makes use of the fact that (Einv⁻¹)_(i,m)=(Einv⁻¹)_(−i,−m).

Eq. 17 becomes

[∂² U _(y)/∂(z′)² ]=[K _(y) ² +B _(reduced) ²(Einv ⁻¹)_(reduced) ^(s) ][S _(x)]  eq. 107

In eqs. 85 and 107 the vectors S_(x) and U_(y) and diagonal matrices K_(y) and K_(x) are trivially reduced to consist of the zeroth and positive terms of the original vectors/matrices. When truncated with truncation order N; the size of the eigen-problems are (N+1)×(N+1) instead of (2N+1)×(2N+1), and require much less computation time to solve.

The solution to the reduced eigen-problems has the same form as eqs. 18-25, but with 4(N+1) coefficients to be determined instead of 4(2N+1). The correct reduced matrices to use in eqs. 22-25 should still be found, so that the reduced form of eq. 15 is satisfied. Here again one could have derived the entire reduced set of eqs. for eq. 15, but it is really only necessary to reduce a few specific terms in order to find A⁻¹, B⁻¹, A⁻¹K_(x), and B⁻¹K_(x)E⁻¹ to use in eqs. 22-25.

Substituting eqs. 18-21 into the second row of eq. 15 gives

W ₂ Q ₂=(K _(x) E ⁻¹ K _(x) −I)V ₂₂  eq. 108

and

(K _(x) E ⁻¹ K _(x) −I)V ₂₁ =K _(x) E ⁻¹ K _(y) W ₁.  eq. 109

Substituting eq. 25 into eq. 108 gives

W₂Q₂=BV₂₂=BB⁻¹W₂Q₂,  eq. 110

which implies that B⁻¹ in eq. 25 should be replaced by the inverse of the reduced matrix B_(reduced) found earlier.

Eqs. 24 and 109 give

$\begin{matrix} {{{BV}_{21} = {{{{BB}^{- 1}\left( \frac{k_{y}}{k_{0}} \right)}K_{x}E^{- 1}W_{1}} = {K_{x}E^{- 1}K_{y}W_{1}}}},} & {{eq}.\mspace{14mu} 111} \end{matrix}$

Which again implies that B⁻¹→(B_(reduced))⁻¹ in eq. 24, and K_(x)E⁻¹ is found by reducing

$\begin{matrix} {\frac{\partial S_{x}}{\partial\left( z^{\prime} \right)} = {\ldots - {K_{x}E^{- 1}K_{y}{U_{x}.}}}} & {{eq}.\mspace{14mu} 112} \end{matrix}$

Since S_(x) is even in x and U_(x) is odd, the reduced matrix for K_(x)E⁻¹ can be found by applying case 3 with ∈=K_(x)E⁻¹:

$\begin{matrix} {{\frac{\partial S_{x\; 0}}{\partial\left( z^{\prime} \right)} = {\ldots - \begin{Bmatrix} {{\left( {K_{x}E^{- 1}} \right)_{0,0}U_{x\; 0}} +} \\ {\sum\limits_{m = 1}^{\infty}{\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{0,m} - \left( {K_{x}E^{- 1}} \right)_{0,{- m}}} \right\rbrack U_{xm}}} \end{Bmatrix}}}{{{{for}\mspace{14mu} i} = 0},{and}}} & {{eq}.\mspace{14mu} 113} \\ {{\frac{\partial S_{xi}}{\partial\left( z^{\prime} \right)} = {\ldots \begin{Bmatrix} {{{\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{i,0} + \left( {K_{x}E^{- 1}} \right)_{{- i},0}} \right)}U_{x\; 0}} +} \\ {\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\begin{bmatrix} {\left( {K_{x}E^{- 1}} \right)_{i,m} +} \\ {\left( {K_{x}E^{- 1}} \right)_{{- i},m} -} \\ {\left( {K_{x}E^{- 1}} \right)_{i,{- m}} -} \\ \left( {K_{x}E^{- 1}} \right)_{{- i},{- m}} \end{bmatrix}U_{xm}}}} \end{Bmatrix}}},{i > 0.}} & {{eq}.\mspace{14mu} 114} \end{matrix}$

This gives

$\begin{matrix} {\left( {K_{x}E^{- 1}} \right)_{reduced}^{s} = \begin{bmatrix} \left( {K_{x}E^{- 1}} \right)_{0,0} & {\left( {K_{x}E^{- 1}} \right)_{0,1} + \left( {K_{x}E^{- 1}} \right)_{0,{- 1}}} & \ldots \\ {\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{1,0} - \left( {K_{x}E^{- 1}} \right)_{{- 1},0}} \right\rbrack} & {\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{1,1} + \left( {K_{x}E^{- 1}} \right)_{{- 1},1} - \left( {K_{x}E^{- 1}} \right)_{1,{- 1}} - \left( {K_{x}E^{- 1}} \right)_{{- 1},{- 1}}} \right\rbrack} & \ldots \\ {\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{2,0} - \left( {K_{x}E^{- 1}} \right)_{{- 2},0}} \right\rbrack} & {\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{2,1} + \left( {K_{x}E^{- 1}} \right)_{{- 2},1} - \left( {K_{x}E^{- 1}} \right)_{2,{- 1}} - \left( {K_{x}E^{- 1}} \right)_{{- 2},{- 1}}} \right\rbrack} & \ldots \\ \vdots & \; & ⋰ \end{bmatrix}} & {{eq}.\mspace{14mu} 115} \end{matrix}$

Substituting eqs. 18-20, 22, and 23 into the fourth row of eq. 15 gives

W ₁ Q ₁=(K _(x) ² −E)V ₁₁ =AV ₁₁ =AA ⁻¹ W ₁ Q ₁  eq. 116

and

(K _(x) ² −E)V ₁₂ =AV ₁₂ =AA ⁻¹ K _(x) K _(y) W ₂.  eq. 117

Since A is replaced by A_(reduced) in eqs. 116 and 117, A⁻¹ should be replaced by (A_(reduced))⁻¹ in both eqs. 22 and 23. K_(x) is simply replaced by a diagonal matrix with the (K_(x))₀₀, (K_(x))₁₁, . . . , (K_(x))_(NN) components of the original K_(x) matrix, as always.

Therefore eqs. 22-25 are replaced by

V ₁₁=(A _(reduced) ^(s))⁻¹ W ₁ Q ₁,  eq. 118

V ₁₂=(k _(y) /k ₀)(A _(reduced) ^(s))⁻¹ K _(x) W ₂,  eq. 119

V ₂₁=(k _(y) /k ₀)(B _(reduced) ^(s))⁻¹(K _(x) E ⁻¹)_(reduced) ^(s) W ₁,  eq. 120

V ₂₂=(B _(reduced) ^(s))⁻¹ W ₂ Q ₂,  eq. 121

where Q₁, W₁, Q₂, and W₂ are the eigenvalue and eigenvector matrices for the new, reduced eigenproblems of eqs. 85 and 107.

This new, reduced eigen-system, combined with the reduced boundary problem, gives exactly the same diffracted amplitudes and diffraction efficiencies as the old formulation for phi=90 for any given truncation order, N, but with much improved computational efficiency. For a given order, N, the computation speed is reduced by a factor of approximately 8 compared to the old formulation.

In some cases, the new, reduced phi=90 algorithms can be significantly faster than even the corresponding classical mount problem with the same polar incidence angle. In the theoretical best case limit, the phi=90 case requires about 62.5% the time as the corresponding classical case. This assumes that the eigen-problem and boundary value problem require equal amounts of time to solve for a given truncation order, N. In practice, this is more or less realized for lower truncation orders. Such a speed advantage can quickly add up when considering the amount of time that may be required to generate a library of several million spectra. In such cases, it may be beneficial to use the phi=90 mount only.

In the other limiting case where a very large truncation order is required, the computation time is basically dominated by the large matrix inversion in the boundary problem (eq. 47). In this limit, the phi=90 case requires approximately 92.5% of the computation time as the phi=0 case. Steps can be taken to make the matrix inversion more efficient, since only the top half is used, which is of some help.

These estimates ignore the fact that there is a little more overhead when constructing the various matrices for the phi=90 case than with the phi=0 case. In practice, the differences in computation speed ranges from being about equal for the phi=90 and phi=0 cases to a 20-30% speed improvement for the phi=90 mount over the corresponding phi=0 case. Either way, the improvement over the old phi=90 formulation is quite significant, and the ideas outlined in the introduction section involving multiple azimuthal datasets can be employed without a disabling increase in computation cost.

To complete the description, the reduced eigen-system is derived for p polarized incident light in the phi=90 conical mount. In this case, the fields satisfy

R _(x,i) =−R _(x,−i)  eq. 122

R_(p,i)=R_(p,−i)  eq. 123

T _(x,i) =−T _(x,−i)  eq. 124

T_(p,i)=T_(p,−i),  eq. 125

in regions I and II, and

S _(x,i) =−S _(x,−i)  eq. 126

S_(y,i)=S_(y,−i)  eq. 127

U_(x,i)=U_(x,−i)  eq. 128

U _(y,i) =−U _(y,−i),  eq. 129

in the grating region.

Eqs. 122-129 applied to the boundary problem lead to the same conclusion as in the s-polarized incidence case, except in this case add the i and −i terms for eqs. 28, 29, 35, and 36, and subtract the −ith from the ith terms in eqs. 26, 27, 33, and 34.

Again, the boundary matching at z=0 and z=d leads to eqs. 26-41 for the boundary equations, but with N+1 sized vectors R_(s), R_(p), T_(s), and T_(p), so long as it is again possible to reduce the eigen-problem as well.

To do this, start with eq. 16 and apply the case 1 reduction:

$\begin{matrix} {\frac{\partial^{2}U_{x\; 0}}{\partial\left( z^{\prime} \right)^{2}} = {{\frac{k_{y}^{2}}{k_{0}^{2}}U_{x\; 0}} - \left\{ {{E_{0,0}U_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\left( {E_{0,m} + E_{0,{- m}}} \right)U_{xm}}}} \right\}}} & {{eq}.\mspace{14mu} 130} \\ {\frac{\partial^{2}U_{xi}}{\partial\left( z^{\prime} \right)^{2}} = {{\frac{k_{y}^{2}}{k_{0}^{2}}U_{xi}} + {\frac{k_{xi}^{2}}{k_{0}^{2}}U_{xi}} - \begin{Bmatrix} {{\frac{1}{2}\left( {E_{i,0} + E_{{- i},0}} \right)U_{x\; 0}} +} \\ {\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\begin{pmatrix} {E_{i,m} +} \\ {E_{{- i},{- m}} +} \\ {E_{{- i},m} +} \\ E_{i,{- m}} \end{pmatrix}U_{xm}}}} \end{Bmatrix}}} & {{eq}.\mspace{14mu} 131} \end{matrix}$

which shows that E_(reduced) is given by

$\begin{matrix} {\mspace{245mu} {{{E_{i,0}U_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\left( {E_{0,m} + E_{0,{- m}}} \right)U_{xm}}}},{i = 0},\mspace{59mu} {and}}} & {{eq}.\mspace{14mu} 132} \\ {\mspace{230mu} {\begin{matrix} {{{\frac{1}{2}\left( {E_{i,0} + E_{{- i},0}} \right)U_{x\; 0}} + {\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\left( {E_{i,m} + E_{{- i},{- m}} + E_{{- i},m} + E_{i,{- m}}} \right)U_{xm}}}}},{i > 0},} & \; \end{matrix}\mspace{59mu} {or}}} & {{eq}.\mspace{14mu} 133} \\ {{E_{reduced}^{p} =  \begin{bmatrix} E_{0,0} & {E_{0,1} + E_{0,{- 1}}} & {E_{0,2} + E_{0,{- 2}}} & \ldots \\ {\frac{1}{2}\left( {E_{1,0} + E_{{- 1},0}} \right)} & {\frac{1}{2}\left( {E_{1,1} + E_{{- 1},{- 1}} + E_{1,{- 1}} + E_{{- 1},1}} \right)} & {\frac{1}{2}\left( {E_{1,2} + E_{{- 1},{- 2}} + E_{1,{- 2}} + E_{{- 1},2}} \right)} & \ldots \\ {\frac{1}{2}\left( {E_{2,0} + E_{{- 2},0}} \right)} & {\frac{1}{2}\left( {E_{2,1} + E_{{- 2},{- 1}} + E_{2,{- 1}} + E_{{- 2},1}} \right)} & {\frac{1}{2}\left( {E_{2,2} + E_{{- 2},{- 2}} + E_{2,{- 2}} + E_{{- 1},2}} \right)} & \ldots \\ \vdots & \; & \; & ⋰ \end{bmatrix}}\mspace{59mu} {Using}} & {{eq}.\mspace{14mu} 134} \\ {\mspace{239mu} {{A_{reduced}^{p} = {K_{x}^{2} - E_{reduced}^{p}}},}} & {{eq}.\mspace{14mu} 135} \end{matrix}$

eq. 16 becomes

[∂² U _(x)/(z′)² ]=[K _(y) ² +A _(reduced) ^(p) ][U _(x)]  eq. 136

where the indices on U_(x) run from 0 to N, and K_(x) and K_(y) are reduced as in the s polarization case.

For eq. 17, case 2 could be applied directly to the product BEinv⁻¹, but a more efficient set of operations is to proceed as in the s polarization case. Einv⁻¹ is reduced by applying case 2 to eq.

$\begin{matrix} {\mspace{149mu} {{\frac{\partial U_{y\; 0}}{\partial z^{\prime}} = {\ldots + {\left( {Einv}^{- 1} \right)_{0,0}S_{x\; 0}}}},\; {i = 0},}\;} & {{eq}.\mspace{14mu} 137} \\ {\mspace{149mu} {{\frac{\partial U_{yi}}{\partial z^{\prime}} = {\ldots + {\sum\limits_{m = 1}^{\infty}{\left\lbrack {\left( {Einv}^{- 1} \right)_{i,m} - \left( {Einv}^{- 1} \right)_{i,{- m}}} \right\rbrack S_{xm}}}}},{i > 0},\mspace{40mu} {or}}} & {{eq}.\mspace{14mu} 138} \\ {\left( {Einv}^{- 1} \right)_{reduced}^{p} = \begin{bmatrix} \left( {Einv}^{- 1} \right)_{0,0} & 0 & 0 & \ldots \\ 0 & {\left( {Einv}^{- 1} \right)_{1,1} + \left( {Einv}^{- 1} \right)_{1,{- 1}}} & {\left( {Einv}^{- 1} \right)_{1,2} + \left( {Einv}^{- 1} \right)_{1,{- 2}}} & \ldots \\ 0 & {\left( {Einv}^{- 1} \right)_{2,1} + \left( {Einv}^{- 1} \right)_{2,{- 1}}} & {\left( {Einv}^{- 1} \right)_{2,2} + \left( {Einv}^{- 1} \right)_{2,{- 2}}} & \ldots \\ \vdots & \; & \; & ⋰ \end{bmatrix}} & {{eq}.\mspace{14mu} 139} \end{matrix}$

where the fact that (Einv⁻¹)_(i,m)=(Einv⁻¹)_(−i,−m) is utilized.

For B, explicitly reduce the product K_(x)E⁻¹K_(x):

$\begin{matrix} {\frac{\partial S_{xi}}{\partial\left( z^{\prime} \right)} = {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}{U_{ym}.}}}}}} & {{eq}.\mspace{14mu} 140} \end{matrix}$

Subtracting the −i th row from the i th row:

$\begin{matrix} {\mspace{239mu} {{\frac{\partial S_{x\; 0}}{\partial\left( z^{\prime} \right)} = {\ldots + 0}},{i = 0},\mspace{59mu} {{{since}\mspace{14mu} k_{x\; 0}} = 0},{and}}} & {{eq}.\mspace{14mu} 141} \\ \begin{matrix} {{2\frac{\partial S_{xi}}{\partial\left( z^{\prime} \right)}} = {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} - {\frac{k_{x - i}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}}}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}\left\lbrack {{\sum\limits_{m = {- \infty}}^{- 1}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack} -}} \\ {{\frac{k_{x - i}}{k_{0}}\left\lbrack {{\sum\limits_{m = {- \infty}}^{- 1}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}\left\lbrack {{- {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,{- m}}\frac{k_{x - m}}{k_{0}}U_{ym}}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack} -}} \\ {{\frac{k_{x - i}}{k_{0}}\left\lbrack {{- {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},{- m}}\frac{k_{x - m}}{k_{0}}U_{ym}}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}\left\lbrack {{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack} -}} \\ {{\frac{k_{x - i}}{k_{0}}\left\lbrack {{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}\left\lbrack {{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack} +}} \\ {{\frac{k_{xi}}{k_{0}}\left\lbrack {{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}\left\lbrack {{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}} +} \right.}}} \\ \left. {{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}} + {\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} \right\rbrack \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = 1}^{\infty}{\left\lbrack {\left( E^{- 1} \right)_{i,{- m}} + \left( E^{- 1} \right)_{i,m} + \left( E^{- 1} \right)_{{- i},{- m}} + \left( E^{- 1} \right)_{{- i},m}} \right\rbrack \frac{k_{xm}}{k_{0}}U_{ym}}}}}} \\ {= {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = 1}^{\infty}{\left\lbrack {{2\left( E^{- 1} \right)_{i,m}} + {2\left( E^{- 1} \right)_{i,{- m}}}} \right\rbrack \frac{k_{xm}}{k_{0}}U_{ym}}}}}} \end{matrix} & \; \\ {\mspace{245mu} {giving}} & \; \\ {{\frac{\partial S_{xi}}{\partial\left( z^{\prime} \right)} = {\ldots + {\frac{k_{xi}}{k_{0}}{\sum\limits_{m = 1}^{\infty}{\left\lbrack {\left( E^{- 1} \right)_{i,m} + \left( E^{- 1} \right)_{i,{- m}}} \right\rbrack \frac{k_{xm}}{k_{0}}U_{ym}}}}}},{i > 0},\mspace{59mu} {{which}\mspace{14mu} {implies}}} & {{eq}.\mspace{14mu} 142} \\ {\left( {K_{x}E^{- 1}K_{x}} \right)_{reduced}^{p} = \begin{bmatrix} 0 & 0 & 0 & \ldots \\ 0 & {{\frac{k_{x\; 1}}{k_{0}}\left\lbrack {\left( E^{- 1} \right)_{1,1} + \left( E^{- 1} \right)_{1,{- 1}}} \right\rbrack}\frac{k_{x\; 1}}{k_{0}}} & {{\frac{k_{x\; 1}}{k_{0}}\left\lbrack {\left( E^{- 1} \right)_{1,2} + \left( E^{- 1} \right)_{1,{- 2}}} \right\rbrack}\frac{k_{x\; 2}}{k_{0}}} & \ldots \\ 0 & {{\frac{k_{x\; 2}}{k_{0}}\left\lbrack {\left( E^{- 1} \right)_{2,1} + \left( E^{- 1} \right)_{2,{- 1}}} \right\rbrack}\frac{k_{x\; 1}}{k_{0}}} & {{\frac{k_{x\; 2}}{k_{0}}\left\lbrack {\left( E^{- 1} \right)_{2,2} + \left( E^{- 1} \right)_{2,{- 2}}} \right\rbrack}\frac{k_{x\; 2}}{k_{0}}} & \ldots \\ \vdots & \; & \; & ⋰ \end{bmatrix}} & {{eq}.\mspace{14mu} 143} \end{matrix}$

in explicit form. Then

B _(reduced) ^(p)=(K _(x) E ⁻¹ K _(x))_(reduced) ^(p) −I  eq. 144

and eq. 17 becomes

[∂² U _(y)/∂(z′)² ]=[K _(y) ² +B _(reduced) ^(p)(Einv ⁻¹)_(reduced) ^(p) ][S _(x)],  eq. 145

where again the indices run from 0 to N and K_(y) is reduced as in the s polarization case.

The corresponding equations to replace eqs 21-25 are found in a similar manner as before. Most of the verification steps are omitted here. A⁻¹ and B⁻¹ are replaced by (A_(reduced))⁻¹ and (B_(reduced)) ⁻¹ as before. To find K_(x)E⁻¹ in eq. 24, use eq. 111 with case 4:

$\begin{matrix} {{\frac{\partial S_{x\; 0}}{\partial\left( z^{\prime} \right)} = {\ldots - \begin{Bmatrix} {{\left( {K_{x}E^{- 1}} \right)_{0,0}U_{x\; 0}} +} \\ {\sum\limits_{m = 1}^{\infty}{\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{0,m} + \left( {K_{x}E^{- 1}} \right)_{0,{- m}}} \right\rbrack U_{xm}}} \end{Bmatrix}}},{i = 0},{and}} & {{eq}.\mspace{14mu} 146} \\ {{\frac{\partial S_{xi}}{\partial\left( z^{\prime} \right)} = {- \begin{Bmatrix} {{{\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{i,0} - \left( {K_{x}E^{- 1}} \right)_{{- i},0}} \right\rbrack}U_{x\; 0}} +} \\ {\frac{1}{2}{\sum\limits_{m = 1}^{\infty}{\begin{bmatrix} {\left( {K_{x}E^{- 1}} \right)_{i,m} + \left( {K_{x}E^{- 1}} \right)_{i,{- m}} -} \\ {\left( {K_{x}E^{- 1}} \right)_{{- i},m} - \left( {K_{x}E^{- 1}} \right)_{{- i},{- m}}} \end{bmatrix}U_{xm}}}} \end{Bmatrix}}},{i < 0.}} & {{eq}.\mspace{14mu} 147} \end{matrix}$

This gives

$\begin{matrix} {\left( {K_{x}E^{- 1}} \right)_{reduced}^{p} = \begin{bmatrix} \left( {K_{x}E^{- 1}} \right)_{0,0} & {\left( {K_{x}E^{- 1}} \right)_{0,1} + \left( {K_{x}E^{- 1}} \right)_{0,{- 1}}} & \ldots \\ {\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{1,0} - \left( {K_{x}E^{- 1}} \right)_{{- 1},0}} \right\rbrack} & {\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{1,1} + \left( {K_{x}E^{- 1}} \right)_{1,{- 1}} - \left( {K_{x}E^{- 1}} \right)_{{- 1},1} - \left( {K_{x}E^{- 1}} \right)_{{- 1},{- 1}}} \right\rbrack} & \ldots \\ {\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{2,0} - \left( {K_{x}E^{- 1}} \right)_{{- 2},0}} \right\rbrack} & {\frac{1}{2}\left\lbrack {\left( {K_{x}E^{- 1}} \right)_{2,1} + \left( {K_{x}E^{- 1}} \right)_{2,{- 1}} - \left( {K_{x}E^{- 1}} \right)_{{- 2},1} - \left( {K_{x}E^{- 1}} \right)_{{- 2},{- 1}}} \right\rbrack} & \ldots \\ \vdots & \; & ⋰ \end{bmatrix}} & {{eq}.\mspace{14mu} 148} \end{matrix}$

Putting all of this together, eqs. 22-25 for p polarized incidence are replaced by

V ₁₁=(A _(reduced) ^(p))⁻¹ W ₁ Q ₁,  eq. 149

V ₁₂=(k _(y) /k ₀)(A _(reduced) ^(p))⁻¹ K _(x) W ₂,  eq. 150

V ₂₁=(k _(y) /k ₀)(B _(reduced) ^(p))⁻¹(K _(x) E ⁻¹)_(reduced) ^(p) W ₁,  eq. 151

V ₂₂=(B _(reduced) ^(p))⁻¹ W ₂ Q ₂,  eq. 152

where Q₁, W₁, Q₂, and W₂ are the eigenvalue and eigenvector matrices for the new, reduced eigen-problems of eqs. 136 and 145. The speed improvement is very similar to the s polarization case.

After solving the reduced boundary problem for the particular s or p incidence case, the diffraction efficiencies can be obtained from

$\begin{matrix} {{DE}_{ri} = {{{R_{s,i}}^{2}{{Re}\left( \frac{k_{I,{zi}}}{k_{0}n_{I}\cos \; \theta} \right)}} + {{R_{p,i}}^{2}{{{Re}\left( \frac{k_{I,{zi}}/n_{I}^{2}}{k_{0}n_{I}\cos \; \theta} \right)}.}}}} & {{eq}.\mspace{14mu} 153} \end{matrix}$

For i=0, eq. 153 is just the specular reflectance for the given incident condition.

It should be pointed out that the only assumption about the grating permittivity expansion coefficients was the symmetry exploited in Eq. 76. In other words, the specific form of the permittivity Fourier coefficients for a binary grating shown in Eq. 9 were not explicitly used in the above descriptions. The grating can consist of more than 2 different materials with differing optical properties, the only difference being that the permittivity Fourier coefficients are different from the coefficients given for the binary structure in Eq. 9. The grating should still satisfy Eq. 76 where required. Additionally, as with the conventional formulation, profile shapes other than rectangular can be treated using a staircase approximation consisting of multiple rectangular grating layers.

The calculated diffraction efficiencies or amplitudes can be used to compute polarized or unpolarized reflectance data, ellipsometric data, or polarimetric data. During an optical grating measurement, one or more datasets are generated by varying the incident wavelength, polar angle of incidence, theta, and rotating the azimuthal angle of incidence between 0 degrees and 90 degrees. The optical data of the one or more datasets are compared to data generated from a theoretical model of the grating using the above calculation methods. A regression analysis is used to optimize the parameters of the theoretical grating model. The result of the optical measurement is given by the optimized grating parameters. The average of the s and p incident calculations can be used to analyze unpolarized reflectance.

The regression algorithm can be the Simplex or Levenberg-Marquardt algorithms, described in W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (2^(nd) Edition), Cambridge University Press, Cambridge, 1992, among others, or can even consist of a simple parameter grid search. The model calculation can be performed in real-time (at the time of measurement), using one or multiple CPUs. The theoretical model spectra can also be pre-calculated ahead of time, generating a library database of spectra, from which the calculation result can be rapidly extracted during the measurement. A neural network can be pre-generated, from which the best-fit model can be directly extracted using a fixed number of relatively simple calculation steps during the measurement.

In many cases, specularly reflected, transmitted, and/or diffracted intensities are detected, and the optical system can be calibrated to give reflectance (OR diffraction efficiency), transmittance, or diffraction efficiency. However, in some cases, particularly for VUV reflectance work, it may be beneficial to normalize some of the intensities with intensities from other structures or from different incidence conditions. These ratios are independent of incident intensity, and a system calibration that involves determining incident intensity may be skipped. The analysis can be done by calculating the corresponding reflectance or diffraction efficiency ratios. For example, a first dataset may be reflected (0 order) intensity 1(0) due to unpolarized light incident at phi=0, and the second dataset may be reflected (0 order) intensity I(90) due to unpolarized light incident at phi=90. The incident intensity will typically not change over short time periods, so if the datasets are collected in close succession, the intensity ratio is the same as the reflectance ratio:

$\begin{matrix} {\frac{I(0)}{I(90)} = \frac{R(0)}{R(90)}} & {{eq}.\mspace{14mu} 154} \end{matrix}$

R(0) and R(90) can be calculated using the conventional phi=0 calculation and new phi=90 calculation presented above. A regression procedure might use the following merit function:

$\begin{matrix} {\chi^{2} = {\sum\limits_{i = 1}^{N}{\left( \frac{1}{\sigma_{i}} \right)^{2}\left( {\left( \frac{R(0)}{R(90)} \right)_{i,{measured}} - \left( \frac{R(0)}{R(90)} \right)_{i,{calculated}}} \right)^{2}}}} & {{eq}.\mspace{14mu} 155} \end{matrix}$

where the subscript i refers to incident condition (usually wavelength), σ_(i) is the estimated uncertainty of the measured reflectance ratio, and N is the total number of data points included for the ratio. The merit function is minimized by the regression procedure, thereby optimizing the grating parameters, which affect the calculated values for both numerator and denominator of the ratio. Note that in this case, the grating parameters are the same for both numerator and denominator.

As describe above the analysis of a diffraction grating problem is of particular use to determining the various characteristics of the diffraction grating structure including, for example, the critical dimensions and the composition of a diffraction grating. The analysis techniques described herein are of particular use in reducing the complexity and increase the speed of such analysis, which is of particular importance in high volume manufacturing processes. It will be recognized that the diffraction problem analysis techniques described herein may be utilized in a wide range of applications where is desirable to analysis a diffraction grating to obtain any of a wide range of types of characteristics of the grating structure.

Further modifications and alternative embodiments of this invention will be apparent to those skilled in the art in view of this description. It will be recognized, therefore, that the present invention is not limited by these example arrangements. Accordingly, this description is to be construed as illustrative only and is for the purpose of teaching those skilled in the art the manner of carrying out the invention. It is to be understood that the forms of the invention herein shown and described are to be taken as the presently preferred embodiments. Various changes may be made in the implementations and architectures. For example, equivalent elements may be substituted for those illustrated and described herein, and certain features of the invention may be utilized independently of the use of other features, all as would be apparent to one skilled in the art after having the benefit of this description of the invention. 

1. A method of measuring characteristics of a diffraction grating structure, comprising: providing incident light comprising one or more multiple wavelengths; providing the incident light in an incident plane of the light that is at a first azimuthal angle of phi=0 with respect to a plane perpendicular to the diffraction grating structure; providing the incident light in the incident plane of the light that is at a second azimuthal angle of phi=90 with respect to a plane perpendicular to the diffraction grating structure; detecting light reflected from the diffraction grating structure when the incident light is at the first azimuthal angle of effectively phi=0 and when the incident light is at the second azimuthal angle of effectively phi=90; utilizing the detected light as part of a diffraction analysis; and determining at least one characteristic of the diffraction grating structure by exploiting symmetry properties of the diffraction analysis such that the computation time for data at the second azimuthal angle is approximately the same as or is less than the computation time for data at the first azimuthal angle.
 2. The method of claim 1, wherein data is only collected at the first azimuthal angle and the second azimuthal angle.
 3. The method of claim 1, wherein the diffraction analysis comprises utilizing a rigorous coupled wave analysis.
 4. The method of claim 3, wherein the symmetry properties comprise symmetry properties of the Fourier expansions of the rigorous coupled wave (RCW) analysis for the second azimuthal angle, allowing RCW eigen- and boundary problems to be reduced in complexity.
 5. The method of claim 4, wherein a second azimuthal angle boundary problem is reduced to a 4(N+1)×4(N+1) system of equations and an eigen-problem is reduced to two (N+1)×(N+1) eigen-systems for a given truncation order, N.
 6. The method of claim 2, wherein four data sets are utilized in the diffraction analysis for each polar angle, the data sets being comprised of two different polarizations at each of the first and second azimuthal angles.
 7. The method of claim 2, further comprising utilizing multiple polar angles at each of the first and second azimuthal angles.
 8. The method of claim 2, wherein the determined characteristic is a geometrical characteristic.
 9. The method of claim 8, wherein the geometrical characteristic is a line width, height and/or depth.
 10. The method of claim 2, wherein computation time for data at the second azimuthal angle is at least 20% less than the computation time for data at the first azimuthal angle.
 11. The method of claim 10, wherein computation time for data at the second azimuthal angle is at least 30% less than the computation time for data at the first azimuthal angle.
 12. The method of claim 1, wherein the computation time for data at the second azimuthal angle is significantly reduced compared to a conventional computation at the second azimuthal angle.
 13. The method of claim 12, wherein the computation time for data at the second azimuthal angle is reduced by a factor of approximately 8 over a conventional calculation at the second azimuthal angle.
 14. The method of claim 1, wherein the incident light comprises multiple wavelengths of light.
 15. A method of characterizing a diffraction grating structure, comprising collecting a first set of reflected data from the grating structure by providing incident light at a first angle of azimuthal incidence with respect to the grating structure; collecting a second set of reflected data from the grating structure by providing incident light at a second angle of azimuthal incidence with respect to the grating structure, the first and second angles being effectively orthogonal and the second angle of azimuthal incidence being different from zero; analyzing a combination of at least the first and second set of reflected data; and utilizing symmetrical characteristics of a diffraction analysis of the second angle of azimuthal incidence reflected data so as to reduce the computation complexity of the analysis of the second set of reflected data during the determination of at least one geometrical characteristic of the grating structure.
 16. The method of claim 15, where one or more of the sets of data are used to normalize other set(s) of data so that optical metrology data comprises ratios of reflected data collected for different incident conditions, avoiding the need to determine incident intensity via an absolute calibration process.
 17. The method of claim 16, wherein the optical metrology data comprises a first ratio of at least a portion of the first set of reflected data and at least a portion of the second set of reflected data.
 18. The method of claim 17, wherein the diffraction analysis comprises a regression or library lookup procedure that minimizes the difference between a calculated reflectance or diffraction efficiency ratio and a measured intensity ratio.
 19. The method of claim 18, wherein an inverse ratio is substituted in specific wavelength regions where the denominator of the first ratio is near zero.
 20. The method of claim 19, wherein a weighting function is used to equalize a contribution to a merit function regardless of a reflectance ratio magnitude.
 21. The method of claim 18, wherein data regions at which a denominator of the first ratio is near zero are dropped from the diffraction analysis.
 22. The method of claim 15, where one or more diffracted orders of reflected data are detected along with or instead of the 0'th order.
 23. The method of claim 15, wherein data is only collected at the first azimuthal angle and the second azimuthal angle.
 24. The method of claim 23, wherein the diffraction analysis comprises utilizing a rigorous coupled wave analysis.
 25. The method of claim 15, wherein four data sets are utilized in the diffraction analysis for each polar angle, the data sets being comprised of two different polarizations at each of the first and second azimuthal angles.
 26. The method of claim 15, further comprising utilizing multiple polar angles at each of the first and second azimuthal angles.
 27. The method of claim 15, wherein the diffraction analysis comprises utilizing a rigorous coupled wave analysis.
 28. The method of claim 27, wherein the symmetry properties comprise symmetry properties of the Fourier expansions of the rigorous coupled wave (RCW) analysis for the second azimuthal angle, allowing RCW eigen- and boundary problems to be reduced in complexity.
 29. The method of claim 28, wherein a second azimuthal angle boundary problem is reduced to a 4(N+1)×4(N+1) system of equations and an eigen-problem is reduced to two (N+1)×(N+1) eigen-systems for a given truncation order, N.
 30. An optical metrology system, comprising: a light source; a sample having a diffraction grating, the light source providing incident light to the diffraction grating, the plane of incidence of the light with respect to the diffraction grating being changeable with respect to at least an azimuthal rotation; a detector collecting data from light diffracted by the diffraction grating, the system being configured to collect data from the diffraction grating when the incident light is at the first azimuthal angle of phi=0 and when the incident light is at the second azimuthal angle of phi=90; a computing system which utilizes the symmetrical characteristics of a diffraction analysis of the second azimuthal angle so as to reduce the computation complexity of an analysis of the data collected from the diffraction grating at the second azimuthal angle during the determination of at least one characteristic of the diffraction grating.
 31. The optical metrology tool of claim 30, wherein the sample is rotated to provide the azimuthal rotation.
 32. The optical metrology tool of claim 30, wherein the plane of incidence of the light is rotated to provide the azimuthal rotation.
 33. The optical metrology tool of claim 30, wherein the polar angle of the incident light is changeable.
 34. The optical metrology tool of claim 30, wherein the light source provides at least VUV wavelengths of light.
 35. The optical metrology tool of claim 30, wherein the incident light is provided through a high numeric aperture optic using an aperture stop to allow incident light at only specific angles.
 36. The optical metrology tool of claim 32, wherein the numeric aperture optic is configured to transmit multiple azimuthal angles of incident light simultaneously.
 37. A method of measuring characteristics of a diffraction grating structure, comprising: providing incident light comprising multiple wavelengths; providing the incident light in an incident plane of the light that is at a first azimuthal angle of effectively phi=90 with respect to a plane perpendicular to the diffraction grating structure; detecting light reflected or diffracted from the diffraction grating structure when the incident light is at the first azimuthal angle; utilizing the detected light as part of a diffraction analysis; and determining at least one characteristic of the diffraction grating structure by exploiting symmetry properties of the diffraction analysis such that the computation time for data is significantly reduced compared to the conventional computation at the first azimuthal angle, and is approximately the same as or is less than the computation time for the comparable classical mount at the first azimuthal angle and a same polar angle.
 38. The method of claim 37, wherein the diffraction analysis comprises utilizing a rigorous coupled wave analysis.
 39. The method of claim 38, wherein the symmetry properties comprise symmetry properties of the Fourier expansions of the rigorous coupled wave (RCW) analysis for the first azimuthal angle case, allowing the RCW eigen- and boundary problems to be reduced in complexity.
 40. The method of claim 39, wherein the first azimuthal angle boundary problem is reduced to a 4(N+1)×4(N+1) system of equations and the eigen-problem is reduced to two (N+1)×(N+1) eigen-systems for a given truncation order, N.
 41. The method of claim 37, wherein two data sets are utilized in the diffraction analysis for each polar angle, the data sets being comprised of two different polarizations at the first azimuthal angle.
 42. The method of claim 37, further comprising utilizing multiple polar angles at the first azimuthal angle.
 43. The method of claim 37, wherein the computation time for data at the first azimuthal angle case is reduced by a factor of approximately 8 over the conventional first azimuthal angle calculation. 